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Theorem preq2 3488
Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
preq2  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)

Proof of Theorem preq2
StepHypRef Expression
1 preq1 3487 . 2  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
2 prcom 3486 . 2  |-  { C ,  A }  =  { A ,  C }
3 prcom 3486 . 2  |-  { C ,  B }  =  { B ,  C }
41, 2, 33eqtr4g 2140 1  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   {cpr 3417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423
This theorem is referenced by:  preq12  3489  preq2i  3491  preq2d  3494  tpeq2  3497  preq12bg  3585  opeq2  3591  uniprg  3636  intprg  3689  prexg  3994  opth  4020  opeqsn  4035  relop  4534  funopg  4984  pr2ne  6572  hashprg  9884  bj-prexg  10969
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