Theorem ssdf

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021)

	Ref	Expression
Hypotheses	ssdf.1	$$⊢Ⅎx\phi$$
	ssdf.2	$$⊢\left(\phi \wedge x\in A\right)\to x\in B$$
Assertion	ssdf	$$⊢\phi \to A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		ssdf.1	$$⊢Ⅎx\phi$$
2		ssdf.2	$$⊢\left(\phi \wedge x\in A\right)\to x\in B$$
3	2	ex	$$⊢\phi \to \left(x\in A\to x\in B\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in Ax\in B$$
5		dfss3	$$⊢A\subseteq B\leftrightarrow \forall x\in Ax\in B$$
6	4 5	sylibr	$$⊢\phi \to A\subseteq B$$