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Theorem opeq2d 3785
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 3779 . 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 1353   <.cop 3595
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-3an 980  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  df-op 3601
This theorem is referenced by:  tfr1onlemaccex  6348  tfrcllemaccex  6361  fundmen  6805  exmidapne  7258  recexnq  7388  suplocexprlemex  7720  elreal2  7828  frecuzrdgrrn  10407  frec2uzrdg  10408  frecuzrdgrcl  10409  frecuzrdgsuc  10413  frecuzrdgrclt  10414  frecuzrdgg  10415  frecuzrdgsuctlem  10422  seqeq2  10448  seqeq3  10449  iseqvalcbv  10456  seq3val  10457  seqvalcd  10458  eucalgval  12053  ennnfonelemp1  12406  ennnfonelemnn0  12422  strsetsid  12494  ressvalsets  12523  strressid  12529  ressinbasd  12532  ressressg  12533  prdsex  12717  imasex  12725  imasival  12726  imasaddvallemg  12735  xpsfval  12766  xpsval  12770  mgpvalg  13131  mgpress  13139  ring1  13234  opprvalg  13239
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