Theorem elintab

Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993)

	Ref	Expression
Hypothesis	inteqab.1	|- A e. _V
Assertion	elintab	|- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )

Proof

Step	Hyp	Ref	Expression
1		inteqab.1	|- A e. _V
2	1	elint	|- ( A e. |^| { x | ph } <-> A. y ( y e. { x | ph } -> A e. y ) )
3		nfsab1	|- F/ x y e. { x | ph }
4		nfv	|- F/ x A e. y
5	3 4	nfim	|- F/ x ( y e. { x | ph } -> A e. y )
6		nfv	|- F/ y ( ph -> A e. x )
7		eleq1w	|- ( y = x -> ( y e. { x | ph } <-> x e. { x | ph } ) )
8		abid	|- ( x e. { x | ph } <-> ph )
9	7 8	syl6bb	|- ( y = x -> ( y e. { x | ph } <-> ph ) )
10		eleq2w	|- ( y = x -> ( A e. y <-> A e. x ) )
11	9 10	imbi12d	|- ( y = x -> ( ( y e. { x | ph } -> A e. y ) <-> ( ph -> A e. x ) ) )
12	5 6 11	cbvalv1	|- ( A. y ( y e. { x | ph } -> A e. y ) <-> A. x ( ph -> A e. x ) )
13	2 12	bitri	|- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )