Theorem opeq1i 3902

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

Hypothesis

Ref	Expression
opeq1i.1	|- A = B

Assertion

Ref	Expression
opeq1i	|- <. A , C >. = <. B , C >.

Proof of Theorem opeq1i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 |- A = B
2		opeq1 3899	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	ax-mp 5	1 |- <. A , C >. = <. B , C >.

Syntax hints:   = 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:  caucvgsrlemfv  8148  caucvgsr  8159  pitonnlem1  8202  axi2m1  8232  axcaucvg  8257  ennnfonelem1  13276  2strstr1g  13453  2strop1g  13455  setsiedg  16207