Theorem syl3anl2

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005) (Proof shortened by Wolf Lammen, 27-Jun-2022)

	Ref	Expression
Hypotheses	syl3anl2.1	$$⊢\phi \to \chi$$
	syl3anl2.2	$$⊢\left(\left(\psi \wedge \chi \wedge \theta \right)\wedge \tau \right)\to \eta$$
Assertion	syl3anl2	$$⊢\left(\left(\psi \wedge \phi \wedge \theta \right)\wedge \tau \right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl3anl2.1	$$⊢\phi \to \chi$$
2		syl3anl2.2	$$⊢\left(\left(\psi \wedge \chi \wedge \theta \right)\wedge \tau \right)\to \eta$$
3	1	3anim2i	$$⊢\left(\psi \wedge \phi \wedge \theta \right)\to \left(\psi \wedge \chi \wedge \theta \right)$$
4	3 2	sylan	$$⊢\left(\left(\psi \wedge \phi \wedge \theta \right)\wedge \tau \right)\to \eta$$