Theorem opeq1d 3862

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 3856	. 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  3865  oteq2  3866  opth  4322  cbvoprab2  6076  djuf1olem  7216  dfplpq2  7537  ltexnqq  7591  nnanq0  7641  addpinq1  7647  prarloclemlo  7677  prarloclem3  7680  prarloclem5  7683  prsrriota  7971  caucvgsrlemfv  7974  caucvgsr  7985  pitonnlem2  8030  pitonn  8031  recidpirq  8041  ax1rid  8060  axrnegex  8062  nntopi  8077  axcaucvglemval  8080  fseq1m1p1  10287  frecuzrdglem  10628  frecuzrdgg  10633  frecuzrdgdomlem  10634  frecuzrdgfunlem  10636  frecuzrdgsuctlem  10640  pfxswrd  11233  swrdccat  11262  swrdccat3blem  11266  fsum2dlemstep  11940  fprod2dlemstep  12128  ennnfonelemp1  12972  ennnfonelemnn0  12988  setscomd  13068  imasaddvallemg  13343