Metamath Proof Explorer


Theorem bj-inrab2

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 ) }

Proof

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 ) }