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Theorem opeq2d 3786
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 3780 . 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 3596
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 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602
This theorem is referenced by:  tfr1onlemaccex  6349  tfrcllemaccex  6362  fundmen  6806  exmidapne  7259  recexnq  7389  suplocexprlemex  7721  elreal2  7829  frecuzrdgrrn  10408  frec2uzrdg  10409  frecuzrdgrcl  10410  frecuzrdgsuc  10414  frecuzrdgrclt  10415  frecuzrdgg  10416  frecuzrdgsuctlem  10423  seqeq2  10449  seqeq3  10450  iseqvalcbv  10457  seq3val  10458  seqvalcd  10459  eucalgval  12054  ennnfonelemp1  12407  ennnfonelemnn0  12423  strsetsid  12495  ressvalsets  12524  strressid  12530  ressinbasd  12533  ressressg  12534  prdsex  12718  imasex  12726  imasival  12727  imasaddvallemg  12736  xpsfval  12767  xpsval  12771  mgpvalg  13133  mgpress  13141  ring1  13236  opprvalg  13241
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