Logiweb aspects of lemma equivalence nec condition0 in pyk

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The predefined "pyk" aspect

define pyk of lemma equivalence nec condition0 as text unicode start of text unicode small l unicode small e unicode small m unicode small m unicode small a unicode space unicode small e unicode small q unicode small u unicode small i unicode small v unicode small a unicode small l unicode small e unicode small n unicode small c unicode small e unicode space unicode small n unicode small e unicode small c unicode space unicode small c unicode small o unicode small n unicode small d unicode small i unicode small t unicode small i unicode small o unicode small n unicode zero unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma equivalence nec condition0 as text unicode start of text unicode capital h unicode small e unicode small l unicode small p unicode small e unicode small r unicode capital e unicode small q unicode capital n unicode small e unicode small c unicode small e unicode small s unicode small s unicode small a unicode small r unicode small y unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma equivalence nec condition0 as system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end define

The user defined "the proof aspect" aspect

define proof of lemma equivalence nec condition0 as lambda var c dot lambda var x dot proof expand quote system zf infer all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer lemma eq subset modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma er is symmetric modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar conclude for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar cut lemma symmetry modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar conclude zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar cut lemma eq subset modus probans quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote modus probans quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote modus ponens metavar var y end metavar zermelo in metavar var big set end metavar modus ponens metavar var x end metavar zermelo in metavar var big set end metavar modus ponens not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar modus ponens zermelo pair zermelo pair metavar var y end metavar comma metavar var y end metavar end pair comma zermelo pair metavar var y end metavar comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar conclude object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set cut lemma set equality suff condition modus ponens object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set modus ponens object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set imply object var var s end var zermelo in the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set conclude the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set cut all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed 1rule deduction modus ponens all metavar var r end metavar indeed all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var big set end metavar indeed quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar infer metavar var y end metavar zermelo in metavar var big set end metavar infer not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar infer zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar infer the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set conclude quote object var var s end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var s end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var t end var end quote avoid zero quote metavar var big set end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var r end metavar end quote endorse quote object var var u end var end quote avoid zero quote metavar var big set end metavar end quote endorse metavar var x end metavar zermelo in metavar var big set end metavar imply metavar var y end metavar zermelo in metavar var big set end metavar imply not0 not0 for all objects object var var s end var indeed object var var s end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var s end var end pair end pair zermelo in metavar var r end metavar imply not0 for all objects object var var s end var indeed for all objects object var var t end var indeed for all objects object var var u end var indeed object var var s end var zermelo in metavar var big set end metavar imply object var var t end var zermelo in metavar var big set end metavar imply object var var u end var zermelo in metavar var big set end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var t end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var t end var comma object var var t end var end pair comma zermelo pair object var var t end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair object var var s end var comma object var var s end var end pair comma zermelo pair object var var s end var comma object var var u end var end pair end pair zermelo in metavar var r end metavar imply zermelo pair zermelo pair metavar var x end metavar comma metavar var x end metavar end pair comma zermelo pair metavar var x end metavar comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar imply the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var x end metavar end pair end pair zermelo in metavar var r end metavar end set zermelo is the set of ph in metavar var big set end metavar such that zermelo pair zermelo pair placeholder-var var a end var comma placeholder-var var a end var end pair comma zermelo pair placeholder-var var a end var comma metavar var y end metavar end pair end pair zermelo in metavar var r end metavar end set end quote state proof state cache var c end expand end define