Theorem pm2.61iii 114

Description: Inference eliminating three antecedents.

Hypotheses

Ref	Expression
pm2.61iii.1	⊢ (¬ φ → (¬ ψ → (¬ χ → θ)))
pm2.61iii.2	⊢ (φ → θ)
pm2.61iii.3	⊢ (ψ → θ)
pm2.61iii.4	⊢ (χ → θ)

Assertion

Ref	Expression
pm2.61iii	⊢ θ

Proof of Theorem pm2.61iii

Step	Hyp	Ref	Expression
1		pm2.61iii.2	. . . . 5 ⊢ (φ → θ)
2	1	a1d 14	. . . 4 ⊢ (φ → (¬ χ → θ))
3	2	a1d 14	. . 3 ⊢ (φ → (¬ ψ → (¬ χ → θ)))
4		pm2.61iii.1	. . 3 ⊢ (¬ φ → (¬ ψ → (¬ χ → θ)))
5	3, 4	pm2.61i 110	. 2 ⊢ (¬ ψ → (¬ χ → θ))
6		pm2.61iii.3	. 2 ⊢ (ψ → θ)
7		pm2.61iii.4	. 2 ⊢ (χ → θ)
8	5, 6, 7	pm2.61ii 113	1 ⊢ θ

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  axrepnd 3740  axacndlem4 3756  axacndlem5 3757  axacnd 3758

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

metamath.org