Theorem opeq12i 2496

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 2493	. 2 |- ((A = B /\ C = D) -> <.A, C>. = <.B, D>.)
4	1, 2, 3	mp2an 699	1 |- <.A, C>. = <.B, D>.

Syntax hints:   = wceq 958  <.cop 2415

This theorem is referenced by:  elxp6 4108  mulidpq 5081  prlem934b 5150  axi2m1 5297  ruclem15 7525  nvop2 8223  nvvop 8224  phop 8473  hhsssh 9134  dedalg 10647  catded 10668

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420

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