Theorem eel00000 42231

Description: Elimination rule similar eel0000 42229, except with five hpothesis steps. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
eel00000.1	⊢ 𝜑
eel00000.2	⊢ 𝜓
eel00000.3	⊢ 𝜒
eel00000.4	⊢ 𝜃
eel00000.5	⊢ 𝜏
eel00000.6	⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel00000	⊢ 𝜂

Proof of Theorem eel00000

Step	Hyp	Ref	Expression
1		eel00000.4	. 2 ⊢ 𝜃
2		eel00000.5	. 2 ⊢ 𝜏
3		eel00000.2	. . 3 ⊢ 𝜓
4		eel00000.3	. . 3 ⊢ 𝜒
5		eel00000.1	. . . 4 ⊢ 𝜑
6		eel00000.6	. . . . 5 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)
7	6	exp41 434	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
8	5, 7	mpan 686	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
9	3, 4, 8	mp2 9	. 2 ⊢ (𝜃 → (𝜏 → 𝜂))
10	1, 2, 9	mp2 9	1 ⊢ 𝜂

Syntax hints:   → wi 4   ∧ wa 395

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 206  df-an 396

This theorem is referenced by: (None)