Theorem eel1111 37968

Description: Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1322 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
eel1111.1	⊢ (𝜑 → 𝜓)
eel1111.2	⊢ (𝜑 → 𝜒)
eel1111.3	⊢ (𝜑 → 𝜃)
eel1111.4	⊢ (𝜑 → 𝜏)
eel1111.5	⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel1111	⊢ (𝜑 → 𝜂)

Proof of Theorem eel1111

Step	Hyp	Ref	Expression
1		eel1111.4	. 2 ⊢ (𝜑 → 𝜏)
2		eel1111.1	. . 3 ⊢ (𝜑 → 𝜓)
3		eel1111.2	. . 3 ⊢ (𝜑 → 𝜒)
4		eel1111.3	. . 3 ⊢ (𝜑 → 𝜃)
5		eel1111.5	. . . 4 ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)
6	5	exp41 636	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))
7	2, 3, 4, 6	syl3c 64	. 2 ⊢ (𝜑 → (𝜏 → 𝜂))
8	1, 7	mpd 15	1 ⊢ (𝜑 → 𝜂)

Syntax hints:   → wi 4   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  sineq0ALT  38195