Theorem eelT12 42282

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

Hypotheses

Ref	Expression
eelT12.1	⊢ (⊤ → 𝜑)
eelT12.2	⊢ (𝜓 → 𝜒)
eelT12.3	⊢ (𝜃 → 𝜏)
eelT12.4	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eelT12	⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Proof of Theorem eelT12

Step	Hyp	Ref	Expression
1		3anass 1093	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) ↔ (⊤ ∧ (𝜓 ∧ 𝜃)))
2		truan 1552	. . 3 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜃)) ↔ (𝜓 ∧ 𝜃))
3	1, 2	bitri 274	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) ↔ (𝜓 ∧ 𝜃))
4		eelT12.3	. . 3 ⊢ (𝜃 → 𝜏)
5		eelT12.2	. . . 4 ⊢ (𝜓 → 𝜒)
6		eelT12.1	. . . . 5 ⊢ (⊤ → 𝜑)
7		eelT12.4	. . . . 5 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
8	6, 7	syl3an1 1161	. . . 4 ⊢ ((⊤ ∧ 𝜒 ∧ 𝜏) → 𝜂)
9	5, 8	syl3an2 1162	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜏) → 𝜂)
10	4, 9	syl3an3 1163	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜂)
11	3, 10	sylbir 234	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ⊤wtru 1542

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 1544

This theorem is referenced by: (None)