Theorem sylanb

Description: A syllogism inference. (Contributed by NM, 18-May-1994)

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

Proof

Step	Hyp	Ref	Expression
1		sylanb.1	⊢ ( 𝜑 ↔ 𝜓 )
2		sylanb.2	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
3	1	biimpi	⊢ ( 𝜑 → 𝜓 )
4	3 2	sylan	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )