Theorem opeq12i 3838

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 3835	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 426	1 |- <. A , C >. = <. B , D >.

Syntax hints:   = wceq 1373   <.cop 3646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652

This theorem is referenced by:  addpinq1  7612  genipv  7657  ltexpri  7761  recexpr  7786  cauappcvgprlemladdru  7804  cauappcvgprlemladdrl  7805  cauappcvgpr  7810  caucvgprlemcl  7824  caucvgprlemladdrl  7826  caucvgpr  7830  caucvgprprlemval  7836  caucvgprprlemnbj  7841  caucvgprprlemmu  7843  caucvgprprlemclphr  7853  caucvgprprlemaddq  7856  caucvgprprlem1  7857  caucvgprprlem2  7858  caucvgsr  7950  pitonnlem1  7993  axi2m1  8023  axcaucvg  8048