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Theorem preq12d 3677
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 3671 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3syl2anc 411 1  |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by:  opeq1  3778  opeq2  3779  xrminrecl  11276  xrminadd  11278  ring1  13189  xmetxp  13900  xmetxpbl  13901  txmetcnp  13911
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