Theorem pm2.85dc 891

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 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))

Proof of Theorem pm2.85dc

Step	Hyp	Ref	Expression
1		df-dc 821	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		orc 702	. . . 4 ⊢ (𝜑 → (𝜑 ∨ (𝜓 → 𝜒)))
3	2	a1d 22	. . 3 ⊢ (𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))
4		olc 701	. . . . . 6 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
5	4	imim1i 60	. . . . 5 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜓 → (𝜑 ∨ 𝜒)))
6		orel1 715	. . . . 5 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜒) → 𝜒))
7	5, 6	syl9r 73	. . . 4 ⊢ (¬ 𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜓 → 𝜒)))
8		olc 701	. . . 4 ⊢ ((𝜓 → 𝜒) → (𝜑 ∨ (𝜓 → 𝜒)))
9	7, 8	syl6 33	. . 3 ⊢ (¬ 𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))
10	3, 9	jaoi 706	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))
11	1, 10	sylbi 120	1 ⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) → (𝜑 ∨ (𝜓 → 𝜒))))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 820

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 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 821

This theorem is referenced by:  orimdidc  892