Theorem mpan

Description: An inference based on modus ponens. (Contributed by NM, 30-Aug-1993) (Proof shortened by Wolf Lammen, 7-Apr-2013)

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

Proof

Step	Hyp	Ref	Expression
1		mpan.1	⊢ 𝜑
2		mpan.2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
3	1	a1i	⊢ ( 𝜓 → 𝜑 )
4	3 2	mpancom	⊢ ( 𝜓 → 𝜒 )