Theorem eel3132

Description: syl2an with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016)

	Ref	Expression
Hypotheses	eel3132.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
	eel3132.2	$$⊢\left(\theta \wedge \psi \right)\to \tau$$
	eel3132.3	$$⊢\left(\chi \wedge \tau \right)\to \eta$$
Assertion	eel3132	$$⊢\left(\phi \wedge \theta \wedge \psi \right)\to \eta$$

Proof

Step	Hyp	Ref	Expression
1		eel3132.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		eel3132.2	$$⊢\left(\theta \wedge \psi \right)\to \tau$$
3		eel3132.3	$$⊢\left(\chi \wedge \tau \right)\to \eta$$
4	1 2 3	syl2an	$$⊢\left(\left(\phi \wedge \psi \right)\wedge \left(\theta \wedge \psi \right)\right)\to \eta$$
5	4	3impdir	$$⊢\left(\phi \wedge \theta \wedge \psi \right)\to \eta$$