Theorem mpani

Description: An inference based on modus ponens. (Contributed by NM, 10-Apr-1994) (Proof shortened by Wolf Lammen, 19-Nov-2012)

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

Proof

Step	Hyp	Ref	Expression
1		mpani.1	⊢ 𝜓
2		mpani.2	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
3	1	a1i	⊢ ( 𝜑 → 𝜓 )
4	3 2	mpand	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )