Theorem opeq12i 3888

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 3885	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 426	1 |- <. A , C >. = <. B , D >.

Syntax hints:   = wceq 1398   <.cop 3692

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-pr 3696  df-op 3698

This theorem is referenced by:  addpinq1  7779  genipv  7824  ltexpri  7928  recexpr  7953  cauappcvgprlemladdru  7971  cauappcvgprlemladdrl  7972  cauappcvgpr  7977  caucvgprlemcl  7991  caucvgprlemladdrl  7993  caucvgpr  7997  caucvgprprlemval  8003  caucvgprprlemnbj  8008  caucvgprprlemmu  8010  caucvgprprlemclphr  8020  caucvgprprlemaddq  8023  caucvgprprlem1  8024  caucvgprprlem2  8025  caucvgsr  8117  pitonnlem1  8160  axi2m1  8190  axcaucvg  8215  konigsbergvtx  16477  konigsbergiedg  16478