Theorem 3ecase 1286

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 1150	. . 3 ⊢ (φ → (ψ → (χ → θ)))
3		3ecase.1	. . . . 5 ⊢ (¬ φ → θ)
4	3	a1d 22	. . . 4 ⊢ (¬ φ → (χ → θ))
5	4	a1d 22	. . 3 ⊢ (¬ φ → (ψ → (χ → θ)))
6	2, 5	pm2.61i 156	. 2 ⊢ (ψ → (χ → θ))
7		3ecase.2	. 2 ⊢ (¬ ψ → θ)
8		3ecase.3	. 2 ⊢ (¬ χ → θ)
9	6, 7, 8	pm2.61nii 158	1 ⊢ θ

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 934

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-an 360  df-3an 936

This theorem is referenced by: (None)