Theorem sylan9ss

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004) (Proof shortened by Andrew Salmon, 14-Jun-2011)

	Ref	Expression
Hypotheses	sylan9ss.1	$$⊢\phi \to A\subseteq B$$
	sylan9ss.2	$$⊢\psi \to B\subseteq C$$
Assertion	sylan9ss	$$⊢\left(\phi \wedge \psi \right)\to A\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		sylan9ss.1	$$⊢\phi \to A\subseteq B$$
2		sylan9ss.2	$$⊢\psi \to B\subseteq C$$
3		sstr	$$⊢\left(A\subseteq B\wedge B\subseteq C\right)\to A\subseteq C$$
4	1 2 3	syl2an	$$⊢\left(\phi \wedge \psi \right)\to A\subseteq C$$