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Theorem preq12d 3485
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1d.1  |-  ( ph  ->  A  =  B )
preq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
preq12d  |-  ( ph  ->  { A ,  C }  =  { B ,  D } )

Proof of Theorem preq12d
StepHypRef Expression
1 preq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 preq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 preq12 3479 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3syl2anc 403 1  |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   {cpr 3407
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413
This theorem is referenced by:  opeq1  3578  opeq2  3579
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