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Theorem opkeq12d 4067
 Description: Equality deduction for ordered pairs. (The proof was shortened by Andrew Salmon, 29-Jun-2011.) (Contributed by NM, 16-Dec-2006.)
Hypotheses
Ref Expression
opkeq1d.1 (φA = B)
opkeq12d.2 (φC = D)
Assertion
Ref Expression
opkeq12d (φ → ⟪A, C⟫ = ⟪B, D⟫)

Proof of Theorem opkeq12d
StepHypRef Expression
1 opkeq1d.1 . 2 (φA = B)
2 opkeq12d.2 . 2 (φC = D)
3 opkeq12 4061 . 2 ((A = B C = D) → ⟪A, C⟫ = ⟪B, D⟫)
41, 2, 3syl2anc 642 1 (φ → ⟪A, C⟫ = ⟪B, D⟫)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = 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:  vfintle  4546
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