Description: Shorter proof of inrab . (Contributed by BJ, 21-Apr-2019) (Proof modification is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | bj-inrab2 | |- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. A | ( ph /\ ps ) } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-inrab | |- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. ( A i^i A ) | ( ph /\ ps ) } |
|
2 | nfv | |- F/ x T. |
|
3 | inidm | |- ( A i^i A ) = A |
|
4 | 3 | a1i | |- ( T. -> ( A i^i A ) = A ) |
5 | 2 4 | bj-rabeqd | |- ( T. -> { x e. ( A i^i A ) | ( ph /\ ps ) } = { x e. A | ( ph /\ ps ) } ) |
6 | 5 | mptru | |- { x e. ( A i^i A ) | ( ph /\ ps ) } = { x e. A | ( ph /\ ps ) } |
7 | 1 6 | eqtri | |- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. A | ( ph /\ ps ) } |