Theorem eelT1 42328

Description: Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Alan Sare, 23-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eelT1.1	⊢ (⊤ → 𝜑)
eelT1.2	⊢ (𝜓 → 𝜒)
eelT1.3	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
eelT1	⊢ (𝜓 → 𝜃)

Proof of Theorem eelT1

Step	Hyp	Ref	Expression
1		eelT1.1	. . 3 ⊢ (⊤ → 𝜑)
2	1	mptru 1546	. 2 ⊢ 𝜑
3		eelT1.2	. 2 ⊢ (𝜓 → 𝜒)
4		eelT1.3	. 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
5	2, 3, 4	sylancr 587	1 ⊢ (𝜓 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 396  ⊤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-tru 1542

This theorem is referenced by: (None)