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Theorem pm5.62dc 891
Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.62dc (DECID 𝜓 → (((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))

Proof of Theorem pm5.62dc
StepHypRef Expression
1 df-dc 781 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2 ordir 766 . . . 4 (((𝜑𝜓) ∨ ¬ 𝜓) ↔ ((𝜑 ∨ ¬ 𝜓) ∧ (𝜓 ∨ ¬ 𝜓)))
32simplbi 268 . . 3 (((𝜑𝜓) ∨ ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
42simplbi2 377 . . . 4 ((𝜑 ∨ ¬ 𝜓) → ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ 𝜓)))
54com12 30 . . 3 ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ 𝜓)))
63, 5impbid2 141 . 2 ((𝜓 ∨ ¬ 𝜓) → (((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
71, 6sylbi 119 1 (DECID 𝜓 → (((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
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