Theorem sylanblrc

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

	Ref	Expression
Hypotheses	sylanblrc.1	$$⊢\phi \to \psi$$
	sylanblrc.2	$$⊢\chi$$
	sylanblrc.3	$$⊢\theta \leftrightarrow \left(\psi \wedge \chi \right)$$
Assertion	sylanblrc	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		sylanblrc.1	$$⊢\phi \to \psi$$
2		sylanblrc.2	$$⊢\chi$$
3		sylanblrc.3	$$⊢\theta \leftrightarrow \left(\psi \wedge \chi \right)$$
4	2	a1i	$$⊢\phi \to \chi$$
5	1 4 3	sylanbrc	$$⊢\phi \to \theta$$