Theorem ecase2d 1026

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 24	. 2 ⊢ (𝜑 → (𝜏 → 𝜏))
2		ecase2d.1	. . . 4 ⊢ (𝜑 → 𝜓)
3		ecase2d.2	. . . . 5 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒))
4	3	pm2.21d 121	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜏))
5	2, 4	mpand 693	. . 3 ⊢ (𝜑 → (𝜒 → 𝜏))
6		ecase2d.3	. . . . 5 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜃))
7	6	pm2.21d 121	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏))
8	2, 7	mpand 693	. . 3 ⊢ (𝜑 → (𝜃 → 𝜏))
9	5, 8	jaod 855	. 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜃) → 𝜏))
10		ecase2d.4	. 2 ⊢ (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))
11	1, 9, 10	mpjaod 856	1 ⊢ (𝜑 → 𝜏)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843

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 209  df-an 399  df-or 844

This theorem is referenced by: (None)