Theorem eelT11 42327

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

Hypotheses

Ref	Expression
eelT11.1	⊢ (⊤ → 𝜑)
eelT11.2	⊢ (𝜓 → 𝜒)
eelT11.3	⊢ (𝜓 → 𝜃)
eelT11.4	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
eelT11	⊢ (𝜓 → 𝜏)

Proof of Theorem eelT11

Step	Hyp	Ref	Expression
1		3anass 1094	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜓) ↔ (⊤ ∧ (𝜓 ∧ 𝜓)))
2		truan 1550	. . 3 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜓)) ↔ (𝜓 ∧ 𝜓))
3		anidm 565	. . 3 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓)
4	1, 2, 3	3bitri 297	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜓) ↔ 𝜓)
5		eelT11.3	. . 3 ⊢ (𝜓 → 𝜃)
6		eelT11.2	. . . 4 ⊢ (𝜓 → 𝜒)
7		eelT11.1	. . . . 5 ⊢ (⊤ → 𝜑)
8		eelT11.4	. . . . 5 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏)
9	7, 8	syl3an1 1162	. . . 4 ⊢ ((⊤ ∧ 𝜒 ∧ 𝜃) → 𝜏)
10	6, 9	syl3an2 1163	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜏)
11	5, 10	syl3an3 1164	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜓) → 𝜏)
12	4, 11	sylbir 234	1 ⊢ (𝜓 → 𝜏)

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)