Theorem ecase3d 909

Description: Deduction for elimination by cases. (Contributed by NM, 2-May-1996.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
ecase3d.1	⊢ (φ → (ψ → θ))
ecase3d.2	⊢ (φ → (χ → θ))
ecase3d.3	⊢ (φ → (¬ (ψ ∨ χ) → θ))

Assertion

Ref	Expression
ecase3d	⊢ (φ → θ)

Proof of Theorem ecase3d

Step	Hyp	Ref	Expression
1		ecase3d.1	. . 3 ⊢ (φ → (ψ → θ))
2		ecase3d.2	. . 3 ⊢ (φ → (χ → θ))
3	1, 2	jaod 369	. 2 ⊢ (φ → ((ψ ∨ χ) → θ))
4		ecase3d.3	. 2 ⊢ (φ → (¬ (ψ ∨ χ) → θ))
5	3, 4	pm2.61d 150	1 ⊢ (φ → θ)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357

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 177  df-or 359

This theorem is referenced by:  ecased  910