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Theorem opeq2d 3629
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
opeq2d  |-  ( ph  -> 
<. C ,  A >.  = 
<. C ,  B >. )

Proof of Theorem opeq2d
StepHypRef Expression
1 opeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 opeq2 3623 . 2  |-  ( A  =  B  ->  <. C ,  A >.  =  <. C ,  B >. )
31, 2syl 14 1  |-  ( ph  -> 
<. C ,  A >.  = 
<. C ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   <.cop 3449
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-3an 926  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 3003  df-sn 3452  df-pr 3453  df-op 3455
This theorem is referenced by:  tfr1onlemaccex  6113  tfrcllemaccex  6126  fundmen  6521  recexnq  6947  elreal2  7366  frecuzrdgrrn  9811  frec2uzrdg  9812  frecuzrdgrcl  9813  frecuzrdgsuc  9817  frecuzrdgrclt  9818  frecuzrdgg  9819  frecuzrdgsuctlem  9826  iseqeq2  9855  iseqeq3  9856  iseqval  9867  iseqvalcbv  9868  iseqvalt  9869  seq3val  9870  eucalgval  11310  setsssvald  11522  ressid2  11546  ressval2  11547
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