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Theorem preqr2 3531
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
preqr2.1  _V
preqr2.2  _V
Assertion
Ref Expression
preqr2  { C ,  }  { C ,  }

Proof of Theorem preqr2
StepHypRef Expression
1 prcom 3437 . . 3  { C ,  }  { ,  C }
2 prcom 3437 . . 3  { C ,  }  { ,  C }
31, 2eqeq12i 2050 . 2  { C ,  }  { C ,  }  { ,  C }  { ,  C }
4 preqr2.1 . . 3  _V
5 preqr2.2 . . 3  _V
64, 5preqr1 3530 . 2  { ,  C }  { ,  C }
73, 6sylbi 114 1  { C ,  }  { C ,  }
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242   wcel 1390   _Vcvv 2551   {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  preq12b  3532  opth  3965  opthreg  4234
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