Theorem syl2an2r

Description: syl2anr with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016) (Proof shortened by Wolf Lammen, 28-Mar-2022)

	Ref	Expression
Hypotheses	syl2an2r.1	$$⊢\phi \to \psi$$
	syl2an2r.2	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
	syl2an2r.3	$$⊢\left(\psi \wedge \theta \right)\to \tau$$
Assertion	syl2an2r	$$⊢\left(\phi \wedge \chi \right)\to \tau$$

Proof

Step	Hyp	Ref	Expression
1		syl2an2r.1	$$⊢\phi \to \psi$$
2		syl2an2r.2	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
3		syl2an2r.3	$$⊢\left(\psi \wedge \theta \right)\to \tau$$
4	1 3	sylan	$$⊢\left(\phi \wedge \theta \right)\to \tau$$
5	2 4	syldan	$$⊢\left(\phi \wedge \chi \right)\to \tau$$