Theorem syl3anb

Description: A triple syllogism inference. (Contributed by NM, 15-Oct-2005)

	Ref	Expression
Hypotheses	syl3anb.1	$$⊢\phi \leftrightarrow \psi$$
	syl3anb.2	$$⊢\chi \leftrightarrow \theta$$
	syl3anb.3	$$⊢\tau \leftrightarrow \eta$$
	syl3anb.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
Assertion	syl3anb	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$

Proof

Step	Hyp	Ref	Expression
1		syl3anb.1	$$⊢\phi \leftrightarrow \psi$$
2		syl3anb.2	$$⊢\chi \leftrightarrow \theta$$
3		syl3anb.3	$$⊢\tau \leftrightarrow \eta$$
4		syl3anb.4	$$⊢\left(\psi \wedge \theta \wedge \eta \right)\to \zeta$$
5	1 2 3	3anbi123i	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\leftrightarrow \left(\psi \wedge \theta \wedge \eta \right)$$
6	5 4	sylbi	$$⊢\left(\phi \wedge \chi \wedge \tau \right)\to \zeta$$