Theorem ssrdv

Description: Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995)

	Ref	Expression
Hypothesis	ssrdv.1	$$⊢\phi \to \left(x\in A\to x\in B\right)$$
Assertion	ssrdv	$$⊢\phi \to A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		ssrdv.1	$$⊢\phi \to \left(x\in A\to x\in B\right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(x\in A\to x\in B\right)$$
3		df-ss	$$⊢A\subseteq B\leftrightarrow \forall x\left(x\in A\to x\in B\right)$$
4	2 3	sylibr	$$⊢\phi \to A\subseteq B$$