Theorem eelT00 42325

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

Hypotheses

Ref	Expression
eelT00.1	⊢ (⊤ → 𝜑)
eelT00.2	⊢ 𝜓
eelT00.3	⊢ 𝜒
eelT00.4	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
eelT00	⊢ 𝜃

Proof of Theorem eelT00

Step	Hyp	Ref	Expression
1		eelT00.3	. 2 ⊢ 𝜒
2		eelT00.2	. . 3 ⊢ 𝜓
3		3anass 1094	. . . . 5 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ (⊤ ∧ (𝜓 ∧ 𝜒)))
4		truan 1550	. . . . 5 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ 𝜒))
5	3, 4	bitri 274	. . . 4 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
6		eelT00.1	. . . . 5 ⊢ (⊤ → 𝜑)
7		eelT00.4	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
8	6, 7	syl3an1 1162	. . . 4 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) → 𝜃)
9	5, 8	sylbir 234	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
10	2, 9	mpan 687	. 2 ⊢ (𝜒 → 𝜃)
11	1, 10	ax-mp 5	1 ⊢ 𝜃

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086  ⊤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 397  df-3an 1088  df-tru 1542

This theorem is referenced by: (None)