Theorem ssrdv

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

	Ref	Expression
Hypothesis	ssrdv.1	|- ( ph -> ( x e. A -> x e. B ) )
Assertion	ssrdv	|- ( ph -> A C_ B )

Proof

Step	Hyp	Ref	Expression
1		ssrdv.1	|- ( ph -> ( x e. A -> x e. B ) )
2	1	alrimiv	|- ( ph -> A. x ( x e. A -> x e. B ) )
3		df-ss	|- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
4	2 3	sylibr	|- ( ph -> A C_ B )