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Theorem opeq1d 3785
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 3779 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2syl 14 1  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  C >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   <.cop 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602
This theorem is referenced by:  oteq1  3788  oteq2  3789  opth  4238  cbvoprab2  5948  djuf1olem  7052  dfplpq2  7353  ltexnqq  7407  nnanq0  7457  addpinq1  7463  prarloclemlo  7493  prarloclem3  7496  prarloclem5  7499  prsrriota  7787  caucvgsrlemfv  7790  caucvgsr  7801  pitonnlem2  7846  pitonn  7847  recidpirq  7857  ax1rid  7876  axrnegex  7878  nntopi  7893  axcaucvglemval  7896  fseq1m1p1  10095  frecuzrdglem  10411  frecuzrdgg  10416  frecuzrdgdomlem  10417  frecuzrdgfunlem  10419  frecuzrdgsuctlem  10423  fsum2dlemstep  11442  fprod2dlemstep  11630  ennnfonelemp1  12407  ennnfonelemnn0  12423  setscomd  12503  imasaddvallemg  12736
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