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Theorem opeq12 3988
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 3986 . 2  |-  ( A  =  C  ->  <. A ,  B >.  =  <. C ,  B >. )
2 opeq2 3987 . 2  |-  ( B  =  D  ->  <. C ,  B >.  =  <. C ,  D >. )
31, 2sylan9eq 2490 1  |-  ( ( A  =  C  /\  B  =  D )  -> 
<. A ,  B >.  = 
<. C ,  D >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653   <.cop 3819
This theorem is referenced by:  opeq12i  3991  opeq12d  3994  cbvopab  4278  opth  4437  copsex2t  4445  copsex2g  4446  relop  5025  funopg  5487  fsn  5908  fnressn  5920  cbvoprab12  6148  eqopi  6385  f1o2ndf1  6456  tposoprab  6517  omeu  6830  brecop  6999  th3q  7015  ecovcom  7017  ecovass  7018  ecovdi  7019  xpf1o  7271  addsrpr  8952  addcnsr  9012  axcnre  9041  seqeq1  11328  fsumcnv  12559  eucalgval2  13074  xpstopnlem1  17843  divstgplem  18152  qqhval2  24368  fprodcnv  25309  brsegle  26044  isrusgra  28430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
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