Theorem eel0001 38945

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
eel0001.1	⊢ 𝜑
eel0001.2	⊢ 𝜓
eel0001.3	⊢ 𝜒
eel0001.4	⊢ (𝜃 → 𝜏)
eel0001.5	⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel0001	⊢ (𝜃 → 𝜂)

Proof of Theorem eel0001

Step	Hyp	Ref	Expression
1		eel0001.3	. 2 ⊢ 𝜒
2		eel0001.4	. 2 ⊢ (𝜃 → 𝜏)
3		eel0001.1	. . 3 ⊢ 𝜑
4		eel0001.2	. . 3 ⊢ 𝜓
5		eel0001.5	. . . 4 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)
6	5	exp41 638	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
7	3, 4, 6	mp2 9	. 2 ⊢ (𝜒 → (𝜏 → 𝜂))
8	1, 2, 7	mpsyl 68	1 ⊢ (𝜃 → 𝜂)

Syntax hints:   → wi 4   ∧ wa 384

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 197  df-an 386

This theorem is referenced by: (None)