Theorem ecase2d 906

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Dec-2012.)

Hypotheses

Ref	Expression
ecase2d.1	⊢ (φ → ψ)
ecase2d.2	⊢ (φ → ¬ (ψ ∧ χ))
ecase2d.3	⊢ (φ → ¬ (ψ ∧ θ))
ecase2d.4	⊢ (φ → (τ ∨ (χ ∨ θ)))

Assertion

Ref	Expression
ecase2d	⊢ (φ → τ)

Proof of Theorem ecase2d

Step	Hyp	Ref	Expression
1		idd 21	. 2 ⊢ (φ → (τ → τ))
2		ecase2d.1	. . . 4 ⊢ (φ → ψ)
3		ecase2d.2	. . . . 5 ⊢ (φ → ¬ (ψ ∧ χ))
4	3	pm2.21d 98	. . . 4 ⊢ (φ → ((ψ ∧ χ) → τ))
5	2, 4	mpand 656	. . 3 ⊢ (φ → (χ → τ))
6		ecase2d.3	. . . . 5 ⊢ (φ → ¬ (ψ ∧ θ))
7	6	pm2.21d 98	. . . 4 ⊢ (φ → ((ψ ∧ θ) → τ))
8	2, 7	mpand 656	. . 3 ⊢ (φ → (θ → τ))
9	5, 8	jaod 369	. 2 ⊢ (φ → ((χ ∨ θ) → τ))
10		ecase2d.4	. 2 ⊢ (φ → (τ ∨ (χ ∨ θ)))
11	1, 9, 10	mpjaod 370	1 ⊢ (φ → τ)

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: (None)