Theorem opeq1d 3578

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 3572	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	syl 14	1 |- ( ph -> <. A , C >. = <. B , C >. )

Syntax hints:   -> wi 4   = wceq 1285   <.cop 3403

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409

This theorem is referenced by:  oteq1  3581  oteq2  3582  opth  3994  cbvoprab2  5602  dfplpq2  6595  ltexnqq  6649  nnanq0  6699  addpinq1  6705  prarloclemlo  6735  prarloclem3  6738  prarloclem5  6741  prsrriota  7015  caucvgsrlemfv  7018  caucvgsr  7029  pitonnlem2  7066  pitonn  7067  recidpirq  7077  ax1rid  7094  axrnegex  7096  nntopi  7111  axcaucvglemval  7114  fseq1m1p1  9177  frecuzrdglem  9482  frecuzrdgg  9487  frecuzrdgdomlem  9488  frecuzrdgfunlem  9490  frecuzrdgsuctlem  9494