Theorem opeq2d 3712

Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)

Hypothesis

Ref	Expression
opeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
opeq2d	|- ( ph -> <. C , A >. = <. C , B >. )

Proof of Theorem opeq2d

Step	Hyp	Ref	Expression
1		opeq1d.1	. 2 |- ( ph -> A = B )
2		opeq2 3706	. 2 |- ( A = B -> <. C , A >. = <. C , B >. )
3	1, 2	syl 14	1 |- ( ph -> <. C , A >. = <. C , B >. )

Syntax hints:   -> wi 4   = wceq 1331   <.cop 3530

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536

This theorem is referenced by:  tfr1onlemaccex  6245  tfrcllemaccex  6258  fundmen  6700  recexnq  7198  suplocexprlemex  7530  elreal2  7638  frecuzrdgrrn  10181  frec2uzrdg  10182  frecuzrdgrcl  10183  frecuzrdgsuc  10187  frecuzrdgrclt  10188  frecuzrdgg  10189  frecuzrdgsuctlem  10196  seqeq2  10222  seqeq3  10223  iseqvalcbv  10230  seq3val  10231  seqvalcd  10232  eucalgval  11735  ennnfonelemp1  11919  ennnfonelemnn0  11935  strsetsid  11992  ressid2  12018  ressval2  12019