This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mpd

Description: A modus ponens deduction. A translation of natural deduction rule -> E ( -> elimination), see natded . Deduction form of ax-mp . Inference associated with a2i . Commuted form of mpcom . (Contributed by NM, 29-Dec-1992)

	Ref	Expression
Hypotheses	mpd.1	$⊢ φ \to ψ$
	mpd.2	$⊢ φ \to (ψ \to χ)$
Assertion	mpd	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		mpd.1	$⊢ φ \to ψ$
2		mpd.2	$⊢ φ \to (ψ \to χ)$
3	2	a2i	$⊢ (φ \to ψ) \to (φ \to χ)$
4	1 3	ax-mp	$⊢ φ \to χ$