Theorem pm2.61iii 132

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 12	. . . 4 ⊢ (φ → (¬ χ → θ))
3	2	a1d 12	. . 3 ⊢ (φ → (¬ ψ → (¬ χ → θ)))
4		pm2.61iii.1	. . 3 ⊢ (¬ φ → (¬ ψ → (¬ χ → θ)))
5	3, 4	pm2.61i 126	. 2 ⊢ (¬ ψ → (¬ χ → θ))
6		pm2.61iii.3	. 2 ⊢ (ψ → θ)
7		pm2.61iii.4	. 2 ⊢ (χ → θ)
8	5, 6, 7	pm2.61ii 130	1 ⊢ θ

Syntax hints:  ¬ wn 2   → wi 3

This theorem is referenced by:  axrepnd 4918  axacndlem4 4934  axacndlem5 4935  axacnd 4936

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

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