Theorem syl2an

Description: A double syllogism inference. For an implication-only version, see syl2im . (Contributed by NM, 31-Jan-1997)

	Ref	Expression
Hypotheses	syl2an.1	$$⊢\phi \to \psi$$
	syl2an.2	$$⊢\tau \to \chi$$
	syl2an.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
Assertion	syl2an	$$⊢\left(\phi \wedge \tau \right)\to \theta$$

Proof

Step	Hyp	Ref	Expression
1		syl2an.1	$$⊢\phi \to \psi$$
2		syl2an.2	$$⊢\tau \to \chi$$
3		syl2an.3	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
4	1 3	sylan	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
5	2 4	sylan2	$$⊢\left(\phi \wedge \tau \right)\to \theta$$