Theorem ecase3ad 911

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013.)

Hypotheses

Ref	Expression
ecase3ad.1	⊢ (φ → (ψ → θ))
ecase3ad.2	⊢ (φ → (χ → θ))
ecase3ad.3	⊢ (φ → ((¬ ψ ∧ ¬ χ) → θ))

Assertion

Ref	Expression
ecase3ad	⊢ (φ → θ)

Proof of Theorem ecase3ad

Step	Hyp	Ref	Expression
1		notnot2 104	. . 3 ⊢ (¬ ¬ ψ → ψ)
2		ecase3ad.1	. . 3 ⊢ (φ → (ψ → θ))
3	1, 2	syl5 28	. 2 ⊢ (φ → (¬ ¬ ψ → θ))
4		notnot2 104	. . 3 ⊢ (¬ ¬ χ → χ)
5		ecase3ad.2	. . 3 ⊢ (φ → (χ → θ))
6	4, 5	syl5 28	. 2 ⊢ (φ → (¬ ¬ χ → θ))
7		ecase3ad.3	. 2 ⊢ (φ → ((¬ ψ ∧ ¬ χ) → θ))
8	3, 6, 7	ecased 910	1 ⊢ (φ → θ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by: (None)