Theorem mp3an1i 1270

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 1264	. 2 ⊢ ((χ ∧ θ) → (φ → τ))
5	4	com12 27	1 ⊢ (φ → ((χ ∧ θ) → τ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934

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

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by: (None)