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Theorem pm2.85dc 890
Description: Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
Assertion
Ref Expression
pm2.85dc (DECID 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))

Proof of Theorem pm2.85dc
StepHypRef Expression
1 df-dc 820 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orc 701 . . . 4 (𝜑 → (𝜑 ∨ (𝜓𝜒)))
32a1d 22 . . 3 (𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
4 olc 700 . . . . . 6 (𝜓 → (𝜑𝜓))
54imim1i 60 . . . . 5 (((𝜑𝜓) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒)))
6 orel1 714 . . . . 5 𝜑 → ((𝜑𝜒) → 𝜒))
75, 6syl9r 73 . . . 4 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜓𝜒)))
8 olc 700 . . . 4 ((𝜓𝜒) → (𝜑 ∨ (𝜓𝜒)))
97, 8syl6 33 . . 3 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
103, 9jaoi 705 . 2 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
111, 10sylbi 120 1 (DECID 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 697  DECID wdc 819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-dc 820
This theorem is referenced by:  orimdidc  891
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