Theorem opeq12i 3909

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 3906	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 430	1 |- <. A , C >. = <. B , D >.

This proof depends on syntax axioms:   = wceq 1402   <.cop 3712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718

This theorem is used by:  addpinq1  7832  genipv  7877  ltexpri  7981  recexpr  8006  cauappcvgprlemladdru  8024  cauappcvgprlemladdrl  8025  cauappcvgpr  8030  caucvgprlemcl  8044  caucvgprlemladdrl  8046  caucvgpr  8050  caucvgprprlemval  8056  caucvgprprlemnbj  8061  caucvgprprlemmu  8063  caucvgprprlemclphr  8073  caucvgprprlemaddq  8076  caucvgprprlem1  8077  caucvgprprlem2  8078  caucvgsr  8170  pitonnlem1  8213  axi2m1  8243  axcaucvg  8268  konigsbergvtx  16889  konigsbergiedg  16890