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Theorem prexg 3947
 Description: The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3480, prprc1 3478, and prprc2 3479. (Contributed by Jim Kingdon, 16-Sep-2018.)
Assertion
Ref Expression
prexg

Proof of Theorem prexg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3448 . . . . . 6
21eleq1d 2106 . . . . 5
3 zfpair2 3945 . . . . 5
42, 3vtoclg 2613 . . . 4
5 preq1 3447 . . . . 5
65eleq1d 2106 . . . 4
74, 6syl5ib 143 . . 3
87vtocleg 2624 . 2
98imp 115 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wceq 1243   wcel 1393  cvv 2557  cpr 3376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by:  opexg  3964  tpexg  4179  onun2  4216  acexmidlemv  5510  xrex  8756
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