Theorem ecase13d 34429

Description: Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)

Hypotheses

Ref	Expression
ecase13d.1	⊢ (𝜑 → ¬ 𝜒)
ecase13d.2	⊢ (𝜑 → ¬ 𝜃)
ecase13d.3	⊢ (𝜑 → (𝜒 ∨ 𝜓 ∨ 𝜃))

Assertion

Ref	Expression
ecase13d	⊢ (𝜑 → 𝜓)

Proof of Theorem ecase13d

Step	Hyp	Ref	Expression
1		ecase13d.2	. 2 ⊢ (𝜑 → ¬ 𝜃)
2		ecase13d.1	. . . 4 ⊢ (𝜑 → ¬ 𝜒)
3		ecase13d.3	. . . . 5 ⊢ (𝜑 → (𝜒 ∨ 𝜓 ∨ 𝜃))
4		3orass 1088	. . . . . 6 ⊢ ((𝜒 ∨ 𝜓 ∨ 𝜃) ↔ (𝜒 ∨ (𝜓 ∨ 𝜃)))
5		df-or 844	. . . . . 6 ⊢ ((𝜒 ∨ (𝜓 ∨ 𝜃)) ↔ (¬ 𝜒 → (𝜓 ∨ 𝜃)))
6	4, 5	bitri 274	. . . . 5 ⊢ ((𝜒 ∨ 𝜓 ∨ 𝜃) ↔ (¬ 𝜒 → (𝜓 ∨ 𝜃)))
7	3, 6	sylib 217	. . . 4 ⊢ (𝜑 → (¬ 𝜒 → (𝜓 ∨ 𝜃)))
8	2, 7	mpd 15	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜃))
9		orcom 866	. . . 4 ⊢ ((𝜓 ∨ 𝜃) ↔ (𝜃 ∨ 𝜓))
10		df-or 844	. . . 4 ⊢ ((𝜃 ∨ 𝜓) ↔ (¬ 𝜃 → 𝜓))
11	9, 10	bitri 274	. . 3 ⊢ ((𝜓 ∨ 𝜃) ↔ (¬ 𝜃 → 𝜓))
12	8, 11	sylib 217	. 2 ⊢ (𝜑 → (¬ 𝜃 → 𝜓))
13	1, 12	mpd 15	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843   ∨ w3o 1084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 844  df-3or 1086

This theorem is referenced by:  ivthALT  34451