Theorem eel11111 42296

Description: Five-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl113anc 1380 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
eel11111.1	⊢ (𝜑 → 𝜓)
eel11111.2	⊢ (𝜑 → 𝜒)
eel11111.3	⊢ (𝜑 → 𝜃)
eel11111.4	⊢ (𝜑 → 𝜏)
eel11111.5	⊢ (𝜑 → 𝜂)
eel11111.6	⊢ (((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
eel11111	⊢ (𝜑 → 𝜁)

Proof of Theorem eel11111

Step	Hyp	Ref	Expression
1		eel11111.4	. 2 ⊢ (𝜑 → 𝜏)
2		eel11111.5	. 2 ⊢ (𝜑 → 𝜂)
3		eel11111.1	. . 3 ⊢ (𝜑 → 𝜓)
4		eel11111.2	. . 3 ⊢ (𝜑 → 𝜒)
5		eel11111.3	. . 3 ⊢ (𝜑 → 𝜃)
6		eel11111.6	. . . . 5 ⊢ (((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)
7	6	exp41 434	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → (𝜃 → (𝜏 → (𝜂 → 𝜁))))
8	7	ex 412	. . 3 ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → (𝜂 → 𝜁)))))
9	3, 4, 5, 8	syl3c 66	. 2 ⊢ (𝜑 → (𝜏 → (𝜂 → 𝜁)))
10	1, 2, 9	mp2d 49	1 ⊢ (𝜑 → 𝜁)

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)