Theorem syl3anr3

Description: A syllogism inference. (Contributed by NM, 23-Aug-2007)

	Ref	Expression
Hypotheses	syl3anr3.1	$$⊢\phi \to \tau$$
	syl3anr3.2	$$⊢\left(\chi \wedge \left(\psi \wedge \theta \wedge \tau \right)\right)\to \eta$$
Assertion	syl3anr3	$$⊢\left(\chi \wedge \left(\psi \wedge \theta \wedge \phi \right)\right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl3anr3.1	$$⊢\phi \to \tau$$
2		syl3anr3.2	$$⊢\left(\chi \wedge \left(\psi \wedge \theta \wedge \tau \right)\right)\to \eta$$
3	1	3anim3i	$$⊢\left(\psi \wedge \theta \wedge \phi \right)\to \left(\psi \wedge \theta \wedge \tau \right)$$
4	3 2	sylan2	$$⊢\left(\chi \wedge \left(\psi \wedge \theta \wedge \phi \right)\right)\to \eta$$