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Theorem preq12d 3452
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 3446 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3syl2anc 391 1  |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   {cpr 3373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-sn 3378  df-pr 3379
This theorem is referenced by:  opeq1  3546  opeq2  3547
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