Theorem opeq12i 3798

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 3795	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 426	1 |- <. A , C >. = <. B , D >.

Syntax hints:   = wceq 1364   <.cop 3610

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616

This theorem is referenced by:  addpinq1  7481  genipv  7526  ltexpri  7630  recexpr  7655  cauappcvgprlemladdru  7673  cauappcvgprlemladdrl  7674  cauappcvgpr  7679  caucvgprlemcl  7693  caucvgprlemladdrl  7695  caucvgpr  7699  caucvgprprlemval  7705  caucvgprprlemnbj  7710  caucvgprprlemmu  7712  caucvgprprlemclphr  7722  caucvgprprlemaddq  7725  caucvgprprlem1  7726  caucvgprprlem2  7727  caucvgsr  7819  pitonnlem1  7862  axi2m1  7892  axcaucvg  7917