Theorem ssrind

Description: Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypothesis	ssrind.1	$$⊢\phi \to A\subseteq B$$
Assertion	ssrind	$$⊢\phi \to A\cap C\subseteq B\cap C$$

Proof

Step	Hyp	Ref	Expression
1		ssrind.1	$$⊢\phi \to A\subseteq B$$
2		ssrin	$$⊢A\subseteq B\to A\cap C\subseteq B\cap C$$
3	1 2	syl	$$⊢\phi \to A\cap C\subseteq B\cap C$$