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Theorem pm5.62dc 863
 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 754 . 2 (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
2 ordir 741 . . . 4 (((𝜑𝜓) ∨ ¬ 𝜓) ↔ ((𝜑 ∨ ¬ 𝜓) ∧ (𝜓 ∨ ¬ 𝜓)))
32simplbi 263 . . 3 (((𝜑𝜓) ∨ ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
42simplbi2 371 . . . 4 ((𝜑 ∨ ¬ 𝜓) → ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ 𝜓)))
54com12 30 . . 3 ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ ¬ 𝜓)))
63, 5impbid2 135 . 2 ((𝜓 ∨ ¬ 𝜓) → (((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
71, 6sylbi 118 1 (DECID 𝜓 → (((𝜑𝜓) ∨ ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639  DECID wdc 753 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640 This theorem depends on definitions:  df-bi 114  df-dc 754 This theorem is referenced by: (None)
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