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Theorem opkeq12 4061
 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
StepHypRef Expression
1 opkeq1 4059 . 2 (A = C → ⟪A, B⟫ = ⟪C, B⟫)
2 opkeq2 4060 . 2 (B = D → ⟪C, B⟫ = ⟪C, D⟫)
31, 2sylan9eq 2405 1 ((A = C B = D) → ⟪A, B⟫ = ⟪C, D⟫)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ⟪copk 4057 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by:  opkeq12i  4064  opkeq12d  4067  opkthg  4131  opkabssvvk  4208  sikss1c1c  4267  ins2kss  4279  ins3kss  4280  sfinltfin  4535
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