Theorem mp3an1i 1225

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 27	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 → 𝜏))
4	1, 3	mp3an1 1219	. 2 ⊢ ((𝜒 ∧ 𝜃) → (𝜑 → 𝜏))
5	4	com12 27	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by: (None)