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Theorem pm5.63dc 853
 Description: Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.63dc (DECID 𝜑 → ((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓))))

Proof of Theorem pm5.63dc
StepHypRef Expression
1 df-dc 743 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ordi 729 . . . 4 ((𝜑 ∨ (¬ 𝜑𝜓)) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑𝜓)))
32simplbi2 367 . . 3 ((𝜑 ∨ ¬ 𝜑) → ((𝜑𝜓) → (𝜑 ∨ (¬ 𝜑𝜓))))
41, 3sylbi 114 . 2 (DECID 𝜑 → ((𝜑𝜓) → (𝜑 ∨ (¬ 𝜑𝜓))))
52simprbi 260 . 2 ((𝜑 ∨ (¬ 𝜑𝜓)) → (𝜑𝜓))
64, 5impbid1 130 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by: (None)
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