Theorem 3ecase 1505

Description: Inference for elimination by cases. (Contributed by NM, 13-Jul-2005.)

Hypotheses

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

Assertion

Ref	Expression
3ecase	⊢ 𝜃

Proof of Theorem 3ecase

Step	Hyp	Ref	Expression
1		3ecase.4	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
2	1	3exp 1137	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3		3ecase.1	. . . 4 ⊢ (¬ 𝜑 → 𝜃)
4	3	2a1d 27	. . 3 ⊢ (¬ 𝜑 → (𝜓 → (𝜒 → 𝜃)))
5	2, 4	pm2.61i 184	. 2 ⊢ (𝜓 → (𝜒 → 𝜃))
6		3ecase.2	. 2 ⊢ (¬ 𝜓 → 𝜃)
7		3ecase.3	. 2 ⊢ (¬ 𝜒 → 𝜃)
8	5, 6, 7	pm2.61nii 186	1 ⊢ 𝜃

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ w3a 1103

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

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by: (None)