Theorem eel0TT 42213

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

Hypotheses

Ref	Expression
eel0TT.1	⊢ 𝜑
eel0TT.2	⊢ (⊤ → 𝜓)
eel0TT.3	⊢ (⊤ → 𝜒)
eel0TT.4	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
eel0TT	⊢ 𝜃

Proof of Theorem eel0TT

Step	Hyp	Ref	Expression
1		eel0TT.3	. . 3 ⊢ (⊤ → 𝜒)
2		truan 1550	. . . 4 ⊢ ((⊤ ∧ 𝜒) ↔ 𝜒)
3		eel0TT.2	. . . . 5 ⊢ (⊤ → 𝜓)
4		eel0TT.1	. . . . . 6 ⊢ 𝜑
5		eel0TT.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)
6	4, 5	mp3an1 1446	. . . . 5 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
7	3, 6	sylan 579	. . . 4 ⊢ ((⊤ ∧ 𝜒) → 𝜃)
8	2, 7	sylbir 234	. . 3 ⊢ (𝜒 → 𝜃)
9	1, 8	syl 17	. 2 ⊢ (⊤ → 𝜃)
10	9	mptru 1546	1 ⊢ 𝜃

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)