Theorem ecased 910

Description: Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012.)

Hypotheses

Ref	Expression
ecased.1	⊢ (φ → (¬ ψ → θ))
ecased.2	⊢ (φ → (¬ χ → θ))
ecased.3	⊢ (φ → ((ψ ∧ χ) → θ))

Assertion

Ref	Expression
ecased	⊢ (φ → θ)

Proof of Theorem ecased

Step	Hyp	Ref	Expression
1		ecased.1	. 2 ⊢ (φ → (¬ ψ → θ))
2		ecased.2	. 2 ⊢ (φ → (¬ χ → θ))
3		pm3.11 485	. . 3 ⊢ (¬ (¬ ψ ∨ ¬ χ) → (ψ ∧ χ))
4		ecased.3	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
5	3, 4	syl5 28	. 2 ⊢ (φ → (¬ (¬ ψ ∨ ¬ χ) → θ))
6	1, 2, 5	ecase3d 909	1 ⊢ (φ → θ)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358

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  df-an 360

This theorem is referenced by:  ecase3ad  911