Theorem eel0T1 42005

Description: An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eel0T1.1	⊢ 𝜑
eel0T1.2	⊢ (⊤ → 𝜓)
eel0T1.3	⊢ (𝜒 → 𝜃)
eel0T1.4	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
eel0T1	⊢ (𝜒 → 𝜏)

Proof of Theorem eel0T1

Step	Hyp	Ref	Expression
1		3anass 1097	. . 3 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜒) ↔ (𝜑 ∧ (⊤ ∧ 𝜒)))
2		simpr 488	. . . 4 ⊢ ((𝜑 ∧ (⊤ ∧ 𝜒)) → (⊤ ∧ 𝜒))
3		eel0T1.1	. . . . 5 ⊢ 𝜑
4	3	jctl 527	. . . 4 ⊢ ((⊤ ∧ 𝜒) → (𝜑 ∧ (⊤ ∧ 𝜒)))
5	2, 4	impbii 212	. . 3 ⊢ ((𝜑 ∧ (⊤ ∧ 𝜒)) ↔ (⊤ ∧ 𝜒))
6		truan 1554	. . 3 ⊢ ((⊤ ∧ 𝜒) ↔ 𝜒)
7	1, 5, 6	3bitri 300	. 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜒) ↔ 𝜒)
8		eel0T1.3	. . 3 ⊢ (𝜒 → 𝜃)
9		eel0T1.2	. . . 4 ⊢ (⊤ → 𝜓)
10		eel0T1.4	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
11	9, 10	syl3an2 1166	. . 3 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜃) → 𝜏)
12	8, 11	syl3an3 1167	. 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜒) → 𝜏)
13	7, 12	sylbir 238	1 ⊢ (𝜒 → 𝜏)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089  ⊤wtru 1544

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  df-tru 1546

This theorem is referenced by: (None)