Theorem sylanblc

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

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

Proof

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