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Theorem preq2 3515
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 3514 . 2  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
2 prcom 3513 . 2  |-  { C ,  A }  =  { A ,  C }
3 prcom 3513 . 2  |-  { C ,  B }  =  { B ,  C }
41, 2, 33eqtr4g 2145 1  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   {cpr 3442
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  preq12  3516  preq2i  3518  preq2d  3521  tpeq2  3524  preq12bg  3612  opeq2  3618  uniprg  3663  intprg  3716  prexg  4029  opth  4055  opeqsn  4070  relop  4574  funopg  5034  pr2ne  6799  hashprg  10181  bj-prexg  11459
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