Theorem opkeq12 4062

Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)

Assertion

Ref	Expression
opkeq12	|- ((A = C /\ B = D) -> <<A, B>> = <<C, D>>)

Proof of Theorem opkeq12

Step	Hyp	Ref	Expression
1		opkeq1 4060	. 2 |- (A = C -> <<A, B>> = <<C, B>>)
2		opkeq2 4061	. 2 |- (B = D -> <<C, B>> = <<C, D>>)
3	1, 2	sylan9eq 2405	1 |- ((A = C /\ B = D) -> <<A, B>> = <<C, D>>)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  <<copk 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  opkeq12i  4065  opkeq12d  4068  opkthg  4132  opkabssvvk  4209  sikss1c1c  4268  ins2kss  4280  ins3kss  4281  sfinltfin  4536