Theorem mpan9

Description: Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008) (Proof shortened by Andrew Salmon, 7-May-2011)

	Ref	Expression
Hypotheses	mpan9.1	⊢ ( 𝜑 → 𝜓 )
	mpan9.2	⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
Assertion	mpan9	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		mpan9.1	⊢ ( 𝜑 → 𝜓 )
2		mpan9.2	⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
3	1 2	syl5	⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )
4	3	impcom	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )