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Theorem preq2 3596
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 3595 . 2  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
2 prcom 3594 . 2  |-  { C ,  A }  =  { A ,  C }
3 prcom 3594 . 2  |-  { C ,  B }  =  { B ,  C }
41, 2, 33eqtr4g 2195 1  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   {cpr 3523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529
This theorem is referenced by:  preq12  3597  preq2i  3599  preq2d  3602  tpeq2  3605  preq12bg  3695  opeq2  3701  uniprg  3746  intprg  3799  prexg  4128  opth  4154  opeqsn  4169  relop  4684  funopg  5152  pr2ne  7041  hashprg  10547  bj-prexg  13098
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