Theorem opeq1d 3868

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 3862	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	syl 14	1 |- ( ph -> <. A , C >. = <. B , C >. )

Syntax hints:   -> wi 4   = wceq 1397   <.cop 3672

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678

This theorem is referenced by:  oteq1  3871  oteq2  3872  opth  4329  cbvoprab2  6093  djuf1olem  7251  dfplpq2  7573  ltexnqq  7627  nnanq0  7677  addpinq1  7683  prarloclemlo  7713  prarloclem3  7716  prarloclem5  7719  prsrriota  8007  caucvgsrlemfv  8010  caucvgsr  8021  pitonnlem2  8066  pitonn  8067  recidpirq  8077  ax1rid  8096  axrnegex  8098  nntopi  8113  axcaucvglemval  8116  fseq1m1p1  10329  frecuzrdglem  10672  frecuzrdgg  10677  frecuzrdgdomlem  10678  frecuzrdgfunlem  10680  frecuzrdgsuctlem  10684  pfxswrd  11286  swrdccat  11315  swrdccat3blem  11319  fsum2dlemstep  11994  fprod2dlemstep  12182  ennnfonelemp1  13026  ennnfonelemnn0  13042  setscomd  13122  imasaddvallemg  13397