Theorem mpdan

Description: An inference based on modus ponens. (Contributed by NM, 23-May-1999) (Proof shortened by Wolf Lammen, 22-Nov-2012)

	Ref	Expression
Hypotheses	mpdan.1	⊢ ( 𝜑 → 𝜓 )
	mpdan.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
Assertion	mpdan	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		mpdan.1	⊢ ( 𝜑 → 𝜓 )
2		mpdan.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
3		id	⊢ ( 𝜑 → 𝜑 )
4	3 1 2	syl2anc	⊢ ( 𝜑 → 𝜒 )