Theorem opeq12i 2483

Description: Equality inference for ordered pairs. (The proof was 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 2480	. 2 |- ((A = B /\ C = D) -> <.A, C>. = <.B, D>.)
4	1, 2, 3	mp2an 695	1 |- <.A, C>. = <.B, D>.

Syntax hints:   = wceq 953  <.cop 2401

This theorem is referenced by:  elxp6 4086  mulidpq 5041  prlem934b 5110  axi2m1 5257  ruclem15 7467  nvop2 8165  nvvop 8166  phop 8408  hhsssh 9059  dedalg 10520  catded 10541

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406

Copyright terms: Public domain