Theorem sylan2

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

	Ref	Expression
Hypotheses	sylan2.1	⊢ ( 𝜑 → 𝜒 )
	sylan2.2	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion	sylan2	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		sylan2.1	⊢ ( 𝜑 → 𝜒 )
2		sylan2.2	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
3	1	adantl	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 )
4	3 2	syldan	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜃 )