Theorem opeq12i 3795

Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Hypotheses

Ref	Expression
opeq1i.1	|- A = B
opeq12i.2	|- C = D

Assertion

Ref	Expression
opeq12i	|- <. A , C >. = <. B , D >.

Proof of Theorem opeq12i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 |- A = B
2		opeq12i.2	. 2 |- C = D
3		opeq12 3792	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 426	1 |- <. A , C >. = <. B , D >.

Syntax hints:   = wceq 1363   <.cop 3607

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613

This theorem is referenced by:  addpinq1  7476  genipv  7521  ltexpri  7625  recexpr  7650  cauappcvgprlemladdru  7668  cauappcvgprlemladdrl  7669  cauappcvgpr  7674  caucvgprlemcl  7688  caucvgprlemladdrl  7690  caucvgpr  7694  caucvgprprlemval  7700  caucvgprprlemnbj  7705  caucvgprprlemmu  7707  caucvgprprlemclphr  7717  caucvgprprlemaddq  7720  caucvgprprlem1  7721  caucvgprprlem2  7722  caucvgsr  7814  pitonnlem1  7857  axi2m1  7887  axcaucvg  7912