Theorem sspsstrd

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr . (Contributed by David Moews, 1-May-2017)

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

Proof

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