Theorem eel12131 42006

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

Hypotheses

Ref	Expression
eel12131.1	⊢ (𝜑 → 𝜓)
eel12131.2	⊢ ((𝜑 ∧ 𝜒) → 𝜃)
eel12131.3	⊢ ((𝜑 ∧ 𝜏) → 𝜂)
eel12131.4	⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
eel12131	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Proof of Theorem eel12131

Step	Hyp	Ref	Expression
1		eel12131.3	. . . . 5 ⊢ ((𝜑 ∧ 𝜏) → 𝜂)
2		eel12131.1	. . . . . . . . 9 ⊢ (𝜑 → 𝜓)
3		eel12131.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4		eel12131.4	. . . . . . . . . 10 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
5	4	3exp 1121	. . . . . . . . 9 ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))
6	2, 3, 5	syl2imc 41	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 → (𝜂 → 𝜁)))
7	6	ex 416	. . . . . . 7 ⊢ (𝜑 → (𝜒 → (𝜑 → (𝜂 → 𝜁))))
8	7	pm2.43b 55	. . . . . 6 ⊢ (𝜒 → (𝜑 → (𝜂 → 𝜁)))
9	8	com13 88	. . . . 5 ⊢ (𝜂 → (𝜑 → (𝜒 → 𝜁)))
10	1, 9	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜏) → (𝜑 → (𝜒 → 𝜁)))
11	10	ex 416	. . 3 ⊢ (𝜑 → (𝜏 → (𝜑 → (𝜒 → 𝜁))))
12	11	pm2.43b 55	. 2 ⊢ (𝜏 → (𝜑 → (𝜒 → 𝜁)))
13	12	3imp231 1115	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089

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 210  df-an 400  df-3an 1091

This theorem is referenced by:  isosctrlem1ALT  42227