Theorem opeq1d 3863

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

Step	Hyp	Ref	Expression
1		opeq1d.1	. 2 |- ( ph -> A = B )
2		opeq1 3857	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	syl 14	1 |- ( ph -> <. A , C >. = <. B , C >. )

Syntax hints:   -> wi 4   = wceq 1395   <.cop 3669

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675

This theorem is referenced by:  oteq1  3866  oteq2  3867  opth  4323  cbvoprab2  6083  djuf1olem  7231  dfplpq2  7552  ltexnqq  7606  nnanq0  7656  addpinq1  7662  prarloclemlo  7692  prarloclem3  7695  prarloclem5  7698  prsrriota  7986  caucvgsrlemfv  7989  caucvgsr  8000  pitonnlem2  8045  pitonn  8046  recidpirq  8056  ax1rid  8075  axrnegex  8077  nntopi  8092  axcaucvglemval  8095  fseq1m1p1  10303  frecuzrdglem  10645  frecuzrdgg  10650  frecuzrdgdomlem  10651  frecuzrdgfunlem  10653  frecuzrdgsuctlem  10657  pfxswrd  11253  swrdccat  11282  swrdccat3blem  11286  fsum2dlemstep  11960  fprod2dlemstep  12148  ennnfonelemp1  12992  ennnfonelemnn0  13008  setscomd  13088  imasaddvallemg  13363