Theorem eelTTT 44731

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

Hypotheses

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

Assertion

Ref	Expression
eelTTT	⊢ 𝜃

Proof of Theorem eelTTT

Step	Hyp	Ref	Expression
1		eelTTT.3	. . 3 ⊢ (⊤ → 𝜒)
2		truan 1550	. . . 4 ⊢ ((⊤ ∧ 𝜒) ↔ 𝜒)
3		eelTTT.2	. . . . 5 ⊢ (⊤ → 𝜓)
4		3anass 1094	. . . . . . 7 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ (⊤ ∧ (𝜓 ∧ 𝜒)))
5		truan 1550	. . . . . . 7 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ 𝜒))
6	4, 5	bitri 275	. . . . . 6 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
7		eelTTT.1	. . . . . . 7 ⊢ (⊤ → 𝜑)
8		eelTTT.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
9	7, 8	syl3an1 1163	. . . . . 6 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜒) → 𝜃)
10	6, 9	sylbir 235	. . . . 5 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
11	3, 10	sylan 580	. . . 4 ⊢ ((⊤ ∧ 𝜒) → 𝜃)
12	2, 11	sylbir 235	. . 3 ⊢ (𝜒 → 𝜃)
13	1, 12	syl 17	. 2 ⊢ (⊤ → 𝜃)
14	13	mptru 1546	1 ⊢ 𝜃

Syntax hints:   → wi 4   ∧ wa 395   ∧ 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 207  df-an 396  df-3an 1088  df-tru 1542

This theorem is referenced by: (None)