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Theorem pm2.85dc 849
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 781 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 orc 668 . . . 4 (𝜑 → (𝜑 ∨ (𝜓𝜒)))
32a1d 22 . . 3 (𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
4 olc 667 . . . . . 6 (𝜓 → (𝜑𝜓))
54imim1i 59 . . . . 5 (((𝜑𝜓) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒)))
6 orel1 679 . . . . 5 𝜑 → ((𝜑𝜒) → 𝜒))
75, 6syl9r 72 . . . 4 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜓𝜒)))
8 olc 667 . . . 4 ((𝜓𝜒) → (𝜑 ∨ (𝜓𝜒)))
97, 8syl6 33 . . 3 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
103, 9jaoi 671 . 2 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
111, 10sylbi 119 1 (DECID 𝜑 → (((𝜑𝜓) → (𝜑𝜒)) → (𝜑 ∨ (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by:  orimdidc  850
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