Theorem mpd3an23 1279

Description: An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)

Hypotheses

Ref	Expression
mpd3an23.1	⊢ (φ → ψ)
mpd3an23.2	⊢ (φ → χ)
mpd3an23.3	⊢ ((φ ∧ ψ ∧ χ) → θ)

Assertion

Ref	Expression
mpd3an23	⊢ (φ → θ)

Proof of Theorem mpd3an23

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (φ → φ)
2		mpd3an23.1	. 2 ⊢ (φ → ψ)
3		mpd3an23.2	. 2 ⊢ (φ → χ)
4		mpd3an23.3	. 2 ⊢ ((φ ∧ ψ ∧ χ) → θ)
5	1, 2, 3, 4	syl3anc 1182	1 ⊢ (φ → θ)

This proof depends on syntax axioms:   → wi 4   ∧ 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)