Theorem ecase3ad 1032

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013.) (Proof shortened by Wolf Lammen, 20-Sep-2024.)

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		ecase3ad.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜃))
2	1	imp 406	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
3		ecase3ad.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
4	3	imp 406	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
5		ecase3ad.3	. . 3 ⊢ (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))
6	5	imp 406	. 2 ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)
7	2, 4, 6	pm2.61ddan 810	1 ⊢ (𝜑 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395

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 206  df-an 396

This theorem is referenced by: (None)