Theorem syl3anl

Description: A triple syllogism inference. (Contributed by NM, 24-Dec-2006)

	Ref	Expression
Hypotheses	syl3anl.1	$$⊢\phi \to \psi$$
	syl3anl.2	$$⊢\chi \to \theta$$
	syl3anl.3	$$⊢\tau \to \eta$$
	syl3anl.4	$$⊢\left(\left(\psi \wedge \theta \wedge \eta \right)\wedge \zeta \right)\to \sigma$$
Assertion	syl3anl	$$⊢\left(\left(\phi \wedge \chi \wedge \tau \right)\wedge \zeta \right)\to \sigma$$

Proof

Step	Hyp	Ref	Expression
1		syl3anl.1	$$⊢\phi \to \psi$$
2		syl3anl.2	$$⊢\chi \to \theta$$
3		syl3anl.3	$$⊢\tau \to \eta$$
4		syl3anl.4	$$⊢\left(\left(\psi \wedge \theta \wedge \eta \right)\wedge \zeta \right)\to \sigma$$
5	1 2 3	3anim123i	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \left(\psi \wedge \theta \wedge \eta \right)$$
6	5 4	sylan	$$⊢\left(\left(\phi \wedge \chi \wedge \tau \right)\wedge \zeta \right)\to \sigma$$