Theorem opeq12i 3575

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 3572	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 416	1 |- <. A , C >. = <. B , D >.

Syntax hints:   = wceq 1284   <.cop 3401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407

This theorem is referenced by:  addpinq1  6654  genipv  6699  ltexpri  6803  recexpr  6828  cauappcvgprlemladdru  6846  cauappcvgprlemladdrl  6847  cauappcvgpr  6852  caucvgprlemcl  6866  caucvgprlemladdrl  6868  caucvgpr  6872  caucvgprprlemval  6878  caucvgprprlemnbj  6883  caucvgprprlemmu  6885  caucvgprprlemclphr  6895  caucvgprprlemaddq  6898  caucvgprprlem1  6899  caucvgprprlem2  6900  caucvgsr  6978  pitonnlem1  7013  axi2m1  7041  axcaucvg  7066