Theorem mpsyl4anc 843

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

Hypotheses

Ref	Expression
mpsyl4anc.1	⊢ 𝜑
mpsyl4anc.2	⊢ 𝜓
mpsyl4anc.3	⊢ 𝜒
mpsyl4anc.4	⊢ (𝜃 → 𝜏)
mpsyl4anc.5	⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
mpsyl4anc	⊢ (𝜃 → 𝜂)

Proof of Theorem mpsyl4anc

Step	Hyp	Ref	Expression
1		mpsyl4anc.1	. . 3 ⊢ 𝜑
2	1	a1i 11	. 2 ⊢ (𝜃 → 𝜑)
3		mpsyl4anc.2	. . 3 ⊢ 𝜓
4	3	a1i 11	. 2 ⊢ (𝜃 → 𝜓)
5		mpsyl4anc.3	. . 3 ⊢ 𝜒
6	5	a1i 11	. 2 ⊢ (𝜃 → 𝜒)
7		mpsyl4anc.4	. 2 ⊢ (𝜃 → 𝜏)
8		mpsyl4anc.5	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂)
9	2, 4, 6, 7, 8	syl1111anc 841	1 ⊢ (𝜃 → 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 207  df-an 396

This theorem is referenced by:  inlinecirc02plem  48707