Theorem opeq1d 3905

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 3899	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	syl 14	1 |- ( ph -> <. A , C >. = <. B , C >. )

Syntax hints:   -> wi 4   = wceq 1402   <.cop 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  oteq1  3908  oteq2  3909  opth  4372  cbvoprab2  6151  djuf1olem  7383  dfplpq2  7711  ltexnqq  7765  nnanq0  7815  addpinq1  7821  prarloclemlo  7851  prarloclem3  7854  prarloclem5  7857  prsrriota  8145  caucvgsrlemfv  8148  caucvgsr  8159  pitonnlem2  8204  pitonn  8205  recidpirq  8215  ax1rid  8234  axrnegex  8236  nntopi  8251  axcaucvglemval  8254  fseq1m1p1  10480  frecuzrdglem  10826  frecuzrdgg  10831  frecuzrdgdomlem  10832  frecuzrdgfunlem  10834  frecuzrdgsuctlem  10838  pfxswrd  11456  swrdccat  11485  swrdccat3blem  11489  fsum2dlemstep  12179  fprod2dlemstep  12367  ennnfonelemp1  13275  ennnfonelemnn0  13291  setscomd  13371  imasaddvallemg  13613