Theorem pm2.61iii 187

Description: Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)

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.4	. 2 ⊢ (𝜒 → 𝜃)
2		pm2.61iii.1	. . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 → (¬ 𝜒 → 𝜃)))
3		pm2.61iii.2	. . . 4 ⊢ (𝜑 → 𝜃)
4	3	a1d 26	. . 3 ⊢ (𝜑 → (¬ 𝜒 → 𝜃))
5		pm2.61iii.3	. . . 4 ⊢ (𝜓 → 𝜃)
6	5	a1d 26	. . 3 ⊢ (𝜓 → (¬ 𝜒 → 𝜃))
7	2, 4, 6	pm2.61ii 185	. 2 ⊢ (¬ 𝜒 → 𝜃)
8	1, 7	pm2.61i 184	1 ⊢ 𝜃

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

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

This theorem is used by:  axrepnd  10596  axacndlem4  10612  axacndlem5  10613  axacnd  10614  axtcond  37048