Theorem syl3an

Description: A triple syllogism inference. (Contributed by NM, 13-May-2004)

	Ref	Expression
Hypotheses	syl3an.1	$$⊢\phi \to \psi$$
	syl3an.2	$$⊢\chi \to \theta$$
	syl3an.3	$$⊢\tau \to \eta$$
	syl3an.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
Assertion	syl3an	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$

Proof

Step	Hyp	Ref	Expression
1		syl3an.1	$$⊢\phi \to \psi$$
2		syl3an.2	$$⊢\chi \to \theta$$
3		syl3an.3	$$⊢\tau \to \eta$$
4		syl3an.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
5	1 2 3	3anim123i	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \left(\psi \wedge \theta \wedge \eta \right)$$
6	5 4	syl	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$