Theorem ssdf2

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	ssdf2.p	$$⊢Ⅎx\phi$$
	ssdf2.a	$$⊢\underset{\_}{Ⅎ}xA$$
	ssdf2.b	$$⊢\underset{\_}{Ⅎ}xB$$
	ssdf2.x	$$⊢\left(\phi \wedge x\in A\right)\to x\in B$$
Assertion	ssdf2	$$⊢\phi \to A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		ssdf2.p	$$⊢Ⅎx\phi$$
2		ssdf2.a	$$⊢\underset{\_}{Ⅎ}xA$$
3		ssdf2.b	$$⊢\underset{\_}{Ⅎ}xB$$
4		ssdf2.x	$$⊢\left(\phi \wedge x\in A\right)\to x\in B$$
5	4	ex	$$⊢\phi \to \left(x\in A\to x\in B\right)$$
6	1 2 3 5	ssrd	$$⊢\phi \to A\subseteq B$$