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Theorem opeq1d 4014
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 4008 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2syl 16 1  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  C >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   <.cop 3841
This theorem is referenced by:  oteq1  4017  oteq2  4018  opth  4464  cbvoprab2  6174  unxpdomlem1  7342  mulcanenq  8868  ax1rid  9067  axrnegex  9068  fseq1m1p1  11154  uzrdglem  11328  s2prop  11892  s4prop  11893  fsum2dlem  12585  ruclem1  12861  imasaddvallem  13785  iscatd2  13937  moni  13993  homadmcd  14228  curf1  14353  curf1cl  14356  curf2  14357  hofcl  14387  gsum2d  15577  imasdsf1olem  18434  ovoliunlem1  19429  cxpcn3  20663  nvi  22124  nvop  22197  phop  22350  fprod2dlem  25335  br8  25410  axlowdimlem15  25926  axlowdim  25931  fvtransport  25997  swrdccat  28274  swrdccat3a  28275  swrdccat3blem  28276  2cshw1lem3  28308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847
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