Theorem ecase3ad 986

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		notnotr 125	. . 3 ⊢ (¬ ¬ 𝜓 → 𝜓)
2		ecase3ad.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
3	1, 2	syl5 34	. 2 ⊢ (𝜑 → (¬ ¬ 𝜓 → 𝜃))
4		notnotr 125	. . 3 ⊢ (¬ ¬ 𝜒 → 𝜒)
5		ecase3ad.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
6	4, 5	syl5 34	. 2 ⊢ (𝜑 → (¬ ¬ 𝜒 → 𝜃))
7		ecase3ad.3	. 2 ⊢ (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))
8	3, 6, 7	ecased 985	1 ⊢ (𝜑 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by: (None)