Metamath Proof Explorer


Theorem elintabg

Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020)

Ref Expression
Assertion elintabg
|- ( A e. V -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) )

Proof

Step Hyp Ref Expression
1 elintg
 |-  ( A e. V -> ( A e. |^| { x | ph } <-> A. y e. { x | ph } A e. y ) )
2 eleq2w
 |-  ( y = x -> ( A e. y <-> A e. x ) )
3 2 ralab2
 |-  ( A. y e. { x | ph } A e. y <-> A. x ( ph -> A e. x ) )
4 1 3 bitrdi
 |-  ( A e. V -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) )