Theorem syl3an132

Description: syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016)

	Ref	Expression
Hypotheses	syl3an132.1	$$⊢\phi \to \psi$$
	syl3an132.2	$$⊢\left(\chi \wedge \theta \right)\to \tau$$
	syl3an132.3	$$⊢\left(\psi \wedge \tau \right)\to \eta$$
Assertion	syl3an132	$$⊢\left(\phi \wedge \chi \wedge \theta \right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		syl3an132.1	$$⊢\phi \to \psi$$
2		syl3an132.2	$$⊢\left(\chi \wedge \theta \right)\to \tau$$
3		syl3an132.3	$$⊢\left(\psi \wedge \tau \right)\to \eta$$
4	1 2 3	syl2an	$$⊢\left(\phi \wedge \left(\chi \wedge \theta \right)\right)\to \eta$$
5	4	3impb	$$⊢\left(\phi \wedge \chi \wedge \theta \right)\to \eta$$