Theorem 3ecase 921

Description: Inference for elimination by cases.

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 831	. . 3 ⊢ (φ → (ψ → (χ → θ)))
3		3ecase.1	. . . . 5 ⊢ (¬ φ → θ)
4	3	a1d 12	. . . 4 ⊢ (¬ φ → (χ → θ))
5	4	a1d 12	. . 3 ⊢ (¬ φ → (ψ → (χ → θ)))
6	2, 5	pm2.61i 126	. 2 ⊢ (ψ → (χ → θ))
7		3ecase.2	. 2 ⊢ (¬ ψ → θ)
8		3ecase.3	. 2 ⊢ (¬ χ → θ)
9	6, 7, 8	pm2.61nii 131	1 ⊢ θ

Syntax hints:  ¬ wn 2   → wi 3   ⋀ w3a 774

This theorem is referenced by:  bcpasc 6915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776

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