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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 → ( 𝜓 → 𝜒 ) )
	mpdd.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
Assertion	mpdd	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

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