Theorem syl2anc2

Description: Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023)

	Ref	Expression
Hypotheses	syl2anc2.1	$$⊢\phi \to \psi$$
	syl2anc2.2	$$⊢\psi \to \chi$$
	syl2anc2.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
Assertion	syl2anc2	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		syl2anc2.1	$$⊢\phi \to \psi$$
2		syl2anc2.2	$$⊢\psi \to \chi$$
3		syl2anc2.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
4	1 2	syl	$$⊢\phi \to \chi$$
5	1 4 3	syl2anc	$$⊢\phi \to \theta$$