Theorem eelTT1 45446

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

Hypotheses

Ref	Expression
eelTT1.1	⊢ (⊤ → 𝜑)
eelTT1.2	⊢ (⊤ → 𝜓)
eelTT1.3	⊢ (𝜒 → 𝜃)
eelTT1.4	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
eelTT1	⊢ (𝜒 → 𝜏)

Proof of Theorem eelTT1

Step	Hyp	Ref	Expression
1		3anass 1111	. . 3 ⊢ ((⊤ ∧ ⊤ ∧ 𝜒) ↔ (⊤ ∧ (⊤ ∧ 𝜒)))
2		anabs5 675	. . 3 ⊢ ((⊤ ∧ (⊤ ∧ 𝜒)) ↔ (⊤ ∧ 𝜒))
3		truan 1581	. . 3 ⊢ ((⊤ ∧ 𝜒) ↔ 𝜒)
4	1, 2, 3	3bitri 300	. 2 ⊢ ((⊤ ∧ ⊤ ∧ 𝜒) ↔ 𝜒)
5		eelTT1.3	. . 3 ⊢ (𝜒 → 𝜃)
6		eelTT1.2	. . . 4 ⊢ (⊤ → 𝜓)
7		eelTT1.1	. . . . 5 ⊢ (⊤ → 𝜑)
8		eelTT1.4	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
9	7, 8	syl3an1 1181	. . . 4 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜏)
10	6, 9	syl3an2 1182	. . 3 ⊢ ((⊤ ∧ ⊤ ∧ 𝜃) → 𝜏)
11	5, 10	syl3an3 1183	. 2 ⊢ ((⊤ ∧ ⊤ ∧ 𝜒) → 𝜏)
12	4, 11	sylbir 238	1 ⊢ (𝜒 → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103  ⊤wtru 1571

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573

This theorem is used by: (None)