Metamath Proof Explorer


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 bitrdi
 |-  ( 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 ) )