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Theorem opeq1d 3932
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 3926 . 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 1649   <.cop 3760
This theorem is referenced by:  oteq1  3935  oteq2  3936  opth  4376  cbvoprab2  6084  unxpdomlem1  7249  mulcanenq  8770  ax1rid  8969  axrnegex  8970  fseq1m1p1  11053  uzrdglem  11224  s2prop  11788  s4prop  11789  fsum2dlem  12481  ruclem1  12757  imasaddvallem  13681  iscatd2  13833  moni  13889  homadmcd  14124  curf1  14249  curf1cl  14252  curf2  14253  hofcl  14283  gsum2d  15473  imasdsf1olem  18311  ovoliunlem1  19265  cxpcn3  20499  nvi  21941  nvop  22014  phop  22167  br8  25137  axlowdimlem15  25609  axlowdim  25614  fvtransport  25680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
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