Theorem ssintrab

Description: Subclass of the intersection of a restricted class abstraction. (Contributed by NM, 30-Jan-2015)

	Ref	Expression
Assertion	ssintrab	|- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )

Proof

Step	Hyp	Ref	Expression
1		df-rab	|- { x e. B | ph } = { x | ( x e. B /\ ph ) }
2	1	inteqi	|- |^| { x e. B | ph } = |^| { x | ( x e. B /\ ph ) }
3	2	sseq2i	|- ( A C_ |^| { x e. B | ph } <-> A C_ |^| { x | ( x e. B /\ ph ) } )
4		impexp	|- ( ( ( x e. B /\ ph ) -> A C_ x ) <-> ( x e. B -> ( ph -> A C_ x ) ) )
5	4	albii	|- ( A. x ( ( x e. B /\ ph ) -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
6		ssintab	|- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x ( ( x e. B /\ ph ) -> A C_ x ) )
7		df-ral	|- ( A. x e. B ( ph -> A C_ x ) <-> A. x ( x e. B -> ( ph -> A C_ x ) ) )
8	5 6 7	3bitr4i	|- ( A C_ |^| { x | ( x e. B /\ ph ) } <-> A. x e. B ( ph -> A C_ x ) )
9	3 8	bitri	|- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )