Theorem sylan

Description: A syllogism inference. (Contributed by NM, 21-Apr-1994) (Proof shortened by Wolf Lammen, 22-Nov-2012)

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

Proof

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