Theorem opeq1d 3719

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 3713	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	syl 14	1 |- ( ph -> <. A , C >. = <. B , C >. )

Syntax hints:   -> wi 4   = wceq 1332   <.cop 3535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541

This theorem is referenced by:  oteq1  3722  oteq2  3723  opth  4167  cbvoprab2  5852  djuf1olem  6946  dfplpq2  7186  ltexnqq  7240  nnanq0  7290  addpinq1  7296  prarloclemlo  7326  prarloclem3  7329  prarloclem5  7332  prsrriota  7620  caucvgsrlemfv  7623  caucvgsr  7634  pitonnlem2  7679  pitonn  7680  recidpirq  7690  ax1rid  7709  axrnegex  7711  nntopi  7726  axcaucvglemval  7729  fseq1m1p1  9906  frecuzrdglem  10215  frecuzrdgg  10220  frecuzrdgdomlem  10221  frecuzrdgfunlem  10223  frecuzrdgsuctlem  10227  fsum2dlemstep  11235  ennnfonelemp1  11955  ennnfonelemnn0  11971