Theorem opeq1d 3866

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 3860	. 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 3670

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676

This theorem is referenced by:  oteq1  3869  oteq2  3870  opth  4327  cbvoprab2  6089  djuf1olem  7243  dfplpq2  7564  ltexnqq  7618  nnanq0  7668  addpinq1  7674  prarloclemlo  7704  prarloclem3  7707  prarloclem5  7710  prsrriota  7998  caucvgsrlemfv  8001  caucvgsr  8012  pitonnlem2  8057  pitonn  8058  recidpirq  8068  ax1rid  8087  axrnegex  8089  nntopi  8104  axcaucvglemval  8107  fseq1m1p1  10320  frecuzrdglem  10663  frecuzrdgg  10668  frecuzrdgdomlem  10669  frecuzrdgfunlem  10671  frecuzrdgsuctlem  10675  pfxswrd  11277  swrdccat  11306  swrdccat3blem  11310  fsum2dlemstep  11985  fprod2dlemstep  12173  ennnfonelemp1  13017  ennnfonelemnn0  13033  setscomd  13113  imasaddvallemg  13388