Theorem eelTT 44762

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

Hypotheses

Ref	Expression
eelTT.1	⊢ (⊤ → 𝜑)
eelTT.2	⊢ (⊤ → 𝜓)
eelTT.3	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
eelTT	⊢ 𝜒

Proof of Theorem eelTT

Step	Hyp	Ref	Expression
1		eelTT.2	. . 3 ⊢ (⊤ → 𝜓)
2		truan 1551	. . . 4 ⊢ ((⊤ ∧ 𝜓) ↔ 𝜓)
3		eelTT.1	. . . . 5 ⊢ (⊤ → 𝜑)
4		eelTT.3	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
5	3, 4	sylan 580	. . . 4 ⊢ ((⊤ ∧ 𝜓) → 𝜒)
6	2, 5	sylbir 235	. . 3 ⊢ (𝜓 → 𝜒)
7	1, 6	syl 17	. 2 ⊢ (⊤ → 𝜒)
8	7	mptru 1547	1 ⊢ 𝜒

Syntax hints:   → wi 4   ∧ wa 395  ⊤wtru 1541

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-tru 1543

This theorem is referenced by: (None)