Theorem opeq1i 3768

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 3765	. 2 |- ( A = B -> <. A , C >. = <. B , C >. )
3	1, 2	ax-mp 5	1 |- <. A , C >. = <. B , C >.

Syntax hints:   = wceq 1348   <.cop 3586

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  caucvgsrlemfv  7753  caucvgsr  7764  pitonnlem1  7807  axi2m1  7837  axcaucvg  7862  ennnfonelem1  12362  2strstr1g  12521  2strop1g  12523