Theorem mp3an1i 1456

Description: An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)

Hypotheses

Ref	Expression
mp3an1i.1	⊢ 𝜓
mp3an1i.2	⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
mp3an1i	⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))

Proof of Theorem mp3an1i

Step	Hyp	Ref	Expression
1		mp3an1i.1	. . 3 ⊢ 𝜓
2		mp3an1i.2	. . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏))
3	2	com12 32	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 → 𝜏))
4	1, 3	mp3an1 1450	. 2 ⊢ ((𝜒 ∧ 𝜃) → (𝜑 → 𝜏))
5	4	com12 32	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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-3an 1089

This theorem is referenced by: (None)