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Theorem opeq1d 4163
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 4157 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2syl 16 1  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  C >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   <.cop 3981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982
This theorem is referenced by:  oteq1  4166  oteq2  4167  opth  4664  cbvoprab2  6258  unxpdomlem1  7618  mulcanenq  9230  ax1rid  9429  axrnegex  9430  fseq1m1p1  11636  uzrdglem  11881  swrd0swrd  12457  swrdccat  12486  swrdccat3a  12487  swrdccat3blem  12488  cshw0  12533  cshwmodn  12534  s2prop  12626  s4prop  12627  fsum2dlem  13339  ruclem1  13615  imasaddvallem  14569  iscatd2  14721  moni  14777  homadmcd  15012  curf1  15137  curf1cl  15140  curf2  15141  hofcl  15171  gsum2dlem2  16567  gsum2dOLD  16569  imasdsf1olem  20064  ovoliunlem1  21101  cxpcn3  22302  axlowdimlem15  23337  axlowdim  23342  nvi  24127  nvop  24200  phop  24353  br8d  26076  fgreu  26124  fprod2dlem  27625  br8  27700  fvtransport  28197  mat1dimmul  31026
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