Theorem eelT01 42220

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

Hypotheses

Ref	Expression
eelT01.1	⊢ (⊤ → 𝜑)
eelT01.2	⊢ 𝜓
eelT01.3	⊢ (𝜒 → 𝜃)
eelT01.4	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
eelT01	⊢ (𝜒 → 𝜏)

Proof of Theorem eelT01

Step	Hyp	Ref	Expression
1		3anass 1093	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ (⊤ ∧ (𝜓 ∧ 𝜒)))
2		truan 1550	. . 3 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ 𝜒))
3		simpr 484	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4		eelT01.2	. . . . 5 ⊢ 𝜓
5	4	jctl 523	. . . 4 ⊢ (𝜒 → (𝜓 ∧ 𝜒))
6	3, 5	impbii 208	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜒)
7	1, 2, 6	3bitri 296	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ 𝜒)
8		eelT01.3	. . 3 ⊢ (𝜒 → 𝜃)
9		eelT01.1	. . . 4 ⊢ (⊤ → 𝜑)
10		eelT01.4	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜏)
11	9, 10	syl3an1 1161	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜏)
12	8, 11	syl3an3 1163	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) → 𝜏)
13	7, 12	sylbir 234	1 ⊢ (𝜒 → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ⊤wtru 1540

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  df-3an 1087  df-tru 1542

This theorem is referenced by: (None)