Theorem sylanblrc

Description: Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019)

	Ref	Expression
Hypotheses	sylanblrc.1	⊢ ( 𝜑 → 𝜓 )
	sylanblrc.2	⊢ 𝜒
	sylanblrc.3	⊢ ( 𝜃 ↔ ( 𝜓 ∧ 𝜒 ) )
Assertion	sylanblrc	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		sylanblrc.1	⊢ ( 𝜑 → 𝜓 )
2		sylanblrc.2	⊢ 𝜒
3		sylanblrc.3	⊢ ( 𝜃 ↔ ( 𝜓 ∧ 𝜒 ) )
4	2	a1i	⊢ ( 𝜑 → 𝜒 )
5	1 4 3	sylanbrc	⊢ ( 𝜑 → 𝜃 )