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Metamath Proof Explorer

Theorem mpdd

Description: A nested modus ponens deduction. Double deduction associated with ax-mp . Deduction associated with mpd . (Contributed by NM, 12-Dec-2004)

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

Proof

Step	Hyp	Ref	Expression
1		mpdd.1	$⊢ φ \to (ψ \to χ)$
2		mpdd.2	$⊢ φ \to (ψ \to (χ \to θ))$
3	2	a2d	$⊢ φ \to ((ψ \to χ) \to (ψ \to θ))$
4	1 3	mpd	$⊢ φ \to (ψ \to θ)$