Theorem pm2.85dc 799

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 731	. 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ))
2		orc 620	. . . 4 ⊢ (φ → (φ ∨ (ψ → χ)))
3	2	a1d 22	. . 3 ⊢ (φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ))))
4		olc 619	. . . . . 6 ⊢ (ψ → (φ ∨ ψ))
5	4	imim1i 54	. . . . 5 ⊢ (((φ ∨ ψ) → (φ ∨ χ)) → (ψ → (φ ∨ χ)))
6		orel1 631	. . . . 5 ⊢ (¬ φ → ((φ ∨ χ) → χ))
7	5, 6	syl9r 67	. . . 4 ⊢ (¬ φ → (((φ ∨ ψ) → (φ ∨ χ)) → (ψ → χ)))
8		olc 619	. . . 4 ⊢ ((ψ → χ) → (φ ∨ (ψ → χ)))
9	7, 8	syl6 29	. . 3 ⊢ (¬ φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ))))
10	3, 9	jaoi 623	. 2 ⊢ ((φ ∨ ¬ φ) → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ))))
11	1, 10	sylbi 114	1 ⊢ (DECID φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ))))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 616  DECID wdc 730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 533  ax-io 617

This theorem depends on definitions:  df-bi 110  df-dc 731

This theorem is referenced by:  orimdidc  800