Theorem sylanl2

Description: A syllogism inference. (Contributed by NM, 1-Jan-2005)

	Ref	Expression
Hypotheses	sylanl2.1	$$⊢\phi \to \chi$$
	sylanl2.2	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \theta \right)\to \tau$$
Assertion	sylanl2	$$⊢\left(\left(\psi \wedge \phi \right)\wedge \theta \right)\to \tau$$

Proof

Step	Hyp	Ref	Expression
1		sylanl2.1	$$⊢\phi \to \chi$$
2		sylanl2.2	$$⊢\left(\left(\psi \wedge \chi \right)\wedge \theta \right)\to \tau$$
3	1	adantl	$$⊢\left(\psi \wedge \phi \right)\to \chi$$
4	3 2	syldanl	$$⊢\left(\left(\psi \wedge \phi \right)\wedge \theta \right)\to \tau$$