Theorem sstrd

Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004)

	Ref	Expression
Hypotheses	sstrd.1	$$⊢\phi \to A\subseteq B$$
	sstrd.2	$$⊢\phi \to B\subseteq C$$
Assertion	sstrd	$$⊢\phi \to A\subseteq C$$

Proof

Step	Hyp	Ref	Expression
1		sstrd.1	$$⊢\phi \to A\subseteq B$$
2		sstrd.2	$$⊢\phi \to B\subseteq C$$
3		sstr	$$⊢\left(A\subseteq B\wedge B\subseteq C\right)\to A\subseteq C$$
4	1 2 3	syl2anc	$$⊢\phi \to A\subseteq C$$