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Theorem opeq1d 3784
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
opeq1d  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  C >. )

Proof of Theorem opeq1d
StepHypRef Expression
1 opeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 opeq1 3778 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2syl 14 1  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  C >. )
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:  oteq1  3787  oteq2  3788  opth  4237  cbvoprab2  5947  djuf1olem  7051  dfplpq2  7352  ltexnqq  7406  nnanq0  7456  addpinq1  7462  prarloclemlo  7492  prarloclem3  7495  prarloclem5  7498  prsrriota  7786  caucvgsrlemfv  7789  caucvgsr  7800  pitonnlem2  7845  pitonn  7846  recidpirq  7856  ax1rid  7875  axrnegex  7877  nntopi  7892  axcaucvglemval  7895  fseq1m1p1  10094  frecuzrdglem  10410  frecuzrdgg  10415  frecuzrdgdomlem  10416  frecuzrdgfunlem  10418  frecuzrdgsuctlem  10422  fsum2dlemstep  11441  fprod2dlemstep  11629  ennnfonelemp1  12406  ennnfonelemnn0  12422  setscomd  12502  imasaddvallemg  12735
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