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 ) ) ) |
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 ) ) ) |