Theorem syl3an12

Description: A double syllogism inference. (Contributed by SN, 15-Sep-2024)

	Ref	Expression
Hypotheses	syl3an12.1	$$⊢\phi \to \psi$$
	syl3an12.2	$$⊢\chi \to \theta$$
	syl3an12.s	$$⊢\left(\psi \wedge \theta \wedge \tau \right)\to \eta$$
Assertion	syl3an12	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl3an12.1	$$⊢\phi \to \psi$$
2		syl3an12.2	$$⊢\chi \to \theta$$
3		syl3an12.s	$$⊢\left(\psi \wedge \theta \wedge \tau \right)\to \eta$$
4		id	$$⊢\tau \to \tau$$
5	1 2 4 3	syl3an	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \eta$$