Theorem opeq1i 3865

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 3862	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	ax-mp 5	1 |- <. A , C >. = <. B , C >.

Syntax hints:   = wceq 1397   <.cop 3672

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678

This theorem is referenced by:  caucvgsrlemfv  8010  caucvgsr  8021  pitonnlem1  8064  axi2m1  8094  axcaucvg  8119  ennnfonelem1  13027  2strstr1g  13204  2strop1g  13206  setsiedg  15902