Theorem ecase13d 1502

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.1	. . 3 ⊢ (𝜑 → ¬ 𝜒)
2		ecase13d.3	. . . 4 ⊢ (𝜑 → (𝜒 ∨ 𝜓 ∨ 𝜃))
3		3orass 1106	. . . . 5 ⊢ ((𝜒 ∨ 𝜓 ∨ 𝜃) ↔ (𝜒 ∨ (𝜓 ∨ 𝜃)))
4		df-or 861	. . . . 5 ⊢ ((𝜒 ∨ (𝜓 ∨ 𝜃)) ↔ (¬ 𝜒 → (𝜓 ∨ 𝜃)))
5	3, 4	bitri 278	. . . 4 ⊢ ((𝜒 ∨ 𝜓 ∨ 𝜃) ↔ (¬ 𝜒 → (𝜓 ∨ 𝜃)))
6	2, 5	sylib 221	. . 3 ⊢ (𝜑 → (¬ 𝜒 → (𝜓 ∨ 𝜃)))
7	1, 6	mpd 16	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜃))
8		ecase13d.2	. 2 ⊢ (𝜑 → ¬ 𝜃)
9	7, 8	olcnd 890	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860   ∨ w3o 1102

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 210  df-or 861  df-3or 1104

This theorem is referenced by:  prlnghpg  29174  ivthALT  36824