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Theorem pm5.63dc 946
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 835 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 ordi 816 . . . 4 ((𝜑 ∨ (¬ 𝜑𝜓)) ↔ ((𝜑 ∨ ¬ 𝜑) ∧ (𝜑𝜓)))
32simplbi2 385 . . 3 ((𝜑 ∨ ¬ 𝜑) → ((𝜑𝜓) → (𝜑 ∨ (¬ 𝜑𝜓))))
41, 3sylbi 121 . 2 (DECID 𝜑 → ((𝜑𝜓) → (𝜑 ∨ (¬ 𝜑𝜓))))
52simprbi 275 . 2 ((𝜑 ∨ (¬ 𝜑𝜓)) → (𝜑𝜓))
64, 5impbid1 142 1 (DECID 𝜑 → ((𝜑𝜓) ↔ (𝜑 ∨ (¬ 𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709
This theorem depends on definitions:  df-bi 117  df-dc 835
This theorem is referenced by: (None)
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