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Metamath Proof Explorer

Theorem mp2d

Description: A double modus ponens deduction. Deduction associated with mp2 . (Contributed by NM, 23-May-2013) (Proof shortened by Wolf Lammen, 23-Jul-2013)

	Ref	Expression
Hypotheses	mp2d.1	⊢ ( 𝜑 → 𝜓 )
	mp2d.2	⊢ ( 𝜑 → 𝜒 )
	mp2d.3	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
Assertion	mp2d	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		mp2d.1	⊢ ( 𝜑 → 𝜓 )
2		mp2d.2	⊢ ( 𝜑 → 𝜒 )
3		mp2d.3	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
4	2 3	mpid	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
5	1 4	mpd	⊢ ( 𝜑 → 𝜃 )