Theorem opeq12i 3625

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 3622	. 2 |- ( ( A = B /\ C = D ) -> <. A , C >. = <. B , D >. )
4	1, 2, 3	mp2an 417	1 |- <. A , C >. = <. B , D >.

Syntax hints:   = wceq 1289   <.cop 3447

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453

This theorem is referenced by:  addpinq1  7013  genipv  7058  ltexpri  7162  recexpr  7187  cauappcvgprlemladdru  7205  cauappcvgprlemladdrl  7206  cauappcvgpr  7211  caucvgprlemcl  7225  caucvgprlemladdrl  7227  caucvgpr  7231  caucvgprprlemval  7237  caucvgprprlemnbj  7242  caucvgprprlemmu  7244  caucvgprprlemclphr  7254  caucvgprprlemaddq  7257  caucvgprprlem1  7258  caucvgprprlem2  7259  caucvgsr  7337  pitonnlem1  7372  axi2m1  7400  axcaucvg  7425