Theorem ss2rab

Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999)

	Ref	Expression
Assertion	ss2rab	|- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		df-rab	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab	|- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1 2	sseq12i	|- ( { x e. A | ph } C_ { x e. A | ps } <-> { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } )
4		ss2ab	|- ( { x | ( x e. A /\ ph ) } C_ { x | ( x e. A /\ ps ) } <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
5		df-ral	|- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		imdistan	|- ( ( x e. A -> ( ph -> ps ) ) <-> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
7	6	albii	|- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
8	5 7	bitr2i	|- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) <-> A. x e. A ( ph -> ps ) )
9	3 4 8	3bitri	|- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )