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Theorem opeq1d 3950
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 3944 . 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 3777
This theorem is referenced by:  oteq1  3953  oteq2  3954  opth  4395  cbvoprab2  6104  unxpdomlem1  7272  mulcanenq  8793  ax1rid  8992  axrnegex  8993  fseq1m1p1  11078  uzrdglem  11252  s2prop  11816  s4prop  11817  fsum2dlem  12509  ruclem1  12785  imasaddvallem  13709  iscatd2  13861  moni  13917  homadmcd  14152  curf1  14277  curf1cl  14280  curf2  14281  hofcl  14311  gsum2d  15501  imasdsf1olem  18356  ovoliunlem1  19351  cxpcn3  20585  nvi  22046  nvop  22119  phop  22272  fprod2dlem  25257  br8  25327  axlowdimlem15  25799  axlowdim  25804  fvtransport  25870  swrdccatin12b  28027  swrdccat3a  28030  swrdccat3b  28031
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783
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