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Theorem opeq1d 3802
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 3796 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2syl 15 1  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  C >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   <.cop 3643
This theorem is referenced by:  oteq1  3805  oteq2  3806  opth  4245  cbvoprab2  5919  unxpdomlem1  7067  mulcanenq  8584  ax1rid  8783  axrnegex  8784  fseq1m1p1  10858  uzrdglem  11020  fsum2dlem  12233  ruclem1  12509  imasaddvallem  13431  iscatd2  13583  moni  13639  homadmcd  13874  curf1  13999  curf1cl  14002  curf2  14003  hofcl  14033  gsum2d  15223  imasdsf1olem  17937  ovoliunlem1  18861  cxpcn3  20088  nvi  21170  nvop  21243  phop  21396  isibfm  23593  rrvmbfm  23645  br8  24113  axlowdimlem15  24584  axlowdim  24589  fvtransport  24655  eqvinopb  24965  isded  25736  dedi  25737  iscatOLD  25754  cati  25755  cmp2morpcats  25961  cmpmorass  25966  cmpidmor3  25970  s2prop  28089  s4prop  28090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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