Theorem eelT11 45674

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 1111	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜓) ↔ (⊤ ∧ (𝜓 ∧ 𝜓)))
2		truan 1581	. . 3 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜓)) ↔ (𝜓 ∧ 𝜓))
3		anidm 575	. . 3 ⊢ ((𝜓 ∧ 𝜓) ↔ 𝜓)
4	1, 2, 3	3bitri 300	. 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 1181	. . . 4 ⊢ ((⊤ ∧ 𝜒 ∧ 𝜃) → 𝜏)
10	6, 9	syl3an2 1182	. . 3 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜃) → 𝜏)
11	5, 10	syl3an3 1183	. 2 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜓) → 𝜏)
12	4, 11	sylbir 238	1 ⊢ (𝜓 → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ⊤wtru 1571

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573

This theorem is used by: (None)