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Theorem opeq12 4159
Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
Assertion
Ref Expression
opeq12  |-  ( ( A  =  C  /\  B  =  D )  -> 
<. A ,  B >.  = 
<. C ,  D >. )

Proof of Theorem opeq12
StepHypRef Expression
1 opeq1 4157 . 2  |-  ( A  =  C  ->  <. A ,  B >.  =  <. C ,  B >. )
2 opeq2 4158 . 2  |-  ( B  =  D  ->  <. C ,  B >.  =  <. C ,  D >. )
31, 2sylan9eq 2512 1  |-  ( ( A  =  C  /\  B  =  D )  -> 
<. A ,  B >.  = 
<. C ,  D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = 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:  opeq12i  4162  opeq12d  4165  cbvopab  4458  opth  4664  copsex2t  4676  copsex2g  4677  relop  5088  funopg  5548  fsn  5980  fnressn  5993  fmptsng  5999  cbvoprab12  6259  eqopi  6710  f1o2ndf1  6780  tposoprab  6881  omeu  7124  brecop  7293  th3q  7309  ecovcom  7311  ecovass  7312  ecovdi  7313  xpf1o  7573  addsrpr  9343  addcnsr  9403  axcnre  9432  seqeq1  11910  fsumcnv  13342  eucalgval2  13858  xpstopnlem1  19498  divstgplem  19807  qqhval2  26545  fprodcnv  27628  brsegle  28273  isrusgra  30682  fmptsnd  30860
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