Theorem eel12131 37959

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	. . . . . 6 ⊢ ((𝜑 ∧ 𝜏) → 𝜂)
2		eel12131.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
3		eel12131.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝜓)
4		eel12131.4	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
5	4	3exp 1256	. . . . . . . . . . 11 ⊢ (𝜓 → (𝜃 → (𝜂 → 𝜁)))
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝜃 → (𝜂 → 𝜁)))
7	2, 6	syl5com 31	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜒) → (𝜑 → (𝜂 → 𝜁)))
8	7	ex 449	. . . . . . . 8 ⊢ (𝜑 → (𝜒 → (𝜑 → (𝜂 → 𝜁))))
9	8	pm2.43b 53	. . . . . . 7 ⊢ (𝜒 → (𝜑 → (𝜂 → 𝜁)))
10	9	com13 86	. . . . . 6 ⊢ (𝜂 → (𝜑 → (𝜒 → 𝜁)))
11	1, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜏) → (𝜑 → (𝜒 → 𝜁)))
12	11	ex 449	. . . 4 ⊢ (𝜑 → (𝜏 → (𝜑 → (𝜒 → 𝜁))))
13	12	pm2.43b 53	. . 3 ⊢ (𝜏 → (𝜑 → (𝜒 → 𝜁)))
14	13	com3l 87	. 2 ⊢ (𝜑 → (𝜒 → (𝜏 → 𝜁)))
15	14	3imp 1249	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031

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  df-3an 1033

This theorem is referenced by:  isosctrlem1ALT  38192