Theorem syl3anr2

Description: A syllogism inference. (Contributed by NM, 1-Aug-2007) (Proof shortened by Wolf Lammen, 27-Jun-2022)

	Ref	Expression
Hypotheses	syl3anr2.1	$$⊢\phi \to \theta$$
	syl3anr2.2	$$⊢\left(\chi \wedge \left(\psi \wedge \theta \wedge \tau \right)\right)\to \eta$$
Assertion	syl3anr2	$$⊢\left(\chi \wedge \left(\psi \wedge \phi \wedge \tau \right)\right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl3anr2.1	$$⊢\phi \to \theta$$
2		syl3anr2.2	$$⊢\left(\chi \wedge \left(\psi \wedge \theta \wedge \tau \right)\right)\to \eta$$
3	1	3anim2i	$$⊢\left(\psi \wedge \phi \wedge \tau \right)\to \left(\psi \wedge \theta \wedge \tau \right)$$
4	3 2	sylan2	$$⊢\left(\chi \wedge \left(\psi \wedge \phi \wedge \tau \right)\right)\to \eta$$