Theorem syld3an1

Description: A syllogism inference. (Contributed by NM, 7-Jul-2008) (Proof shortened by Wolf Lammen, 26-Jun-2022)

	Ref	Expression
Hypotheses	syld3an1.1	$$⊢\left(\chi \wedge \psi \wedge \theta \right)\to \phi$$
	syld3an1.2	$$⊢\left(\phi \wedge \psi \wedge \theta \right)\to \tau$$
Assertion	syld3an1	$$⊢\left(\chi \wedge \psi \wedge \theta \right)\to \tau$$

Proof

Step	Hyp	Ref	Expression
1		syld3an1.1	$$⊢\left(\chi \wedge \psi \wedge \theta \right)\to \phi$$
2		syld3an1.2	$$⊢\left(\phi \wedge \psi \wedge \theta \right)\to \tau$$
3		simp2	$$⊢\left(\chi \wedge \psi \wedge \theta \right)\to \psi$$
4		simp3	$$⊢\left(\chi \wedge \psi \wedge \theta \right)\to \theta$$
5	1 3 4 2	syl3anc	$$⊢\left(\chi \wedge \psi \wedge \theta \right)\to \tau$$