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Theorem opeq1d 3711
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 3705 . 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 1331   <.cop 3530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536
This theorem is referenced by:  oteq1  3714  oteq2  3715  opth  4159  cbvoprab2  5844  djuf1olem  6938  dfplpq2  7169  ltexnqq  7223  nnanq0  7273  addpinq1  7279  prarloclemlo  7309  prarloclem3  7312  prarloclem5  7315  prsrriota  7603  caucvgsrlemfv  7606  caucvgsr  7617  pitonnlem2  7662  pitonn  7663  recidpirq  7673  ax1rid  7692  axrnegex  7694  nntopi  7709  axcaucvglemval  7712  fseq1m1p1  9882  frecuzrdglem  10191  frecuzrdgg  10196  frecuzrdgdomlem  10197  frecuzrdgfunlem  10199  frecuzrdgsuctlem  10203  fsum2dlemstep  11210  ennnfonelemp1  11926  ennnfonelemnn0  11942
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