Theorem rabrab

Description: Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017)

	Ref	Expression
Assertion	rabrab	|- { x e. { x e. A | ph } | ps } = { x e. A | ( ph /\ ps ) }

Proof

Step	Hyp	Ref	Expression
1		rabid	|- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
2	1	anbi1i	|- ( ( x e. { x e. A | ph } /\ ps ) <-> ( ( x e. A /\ ph ) /\ ps ) )
3		anass	|- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
4	2 3	bitri	|- ( ( x e. { x e. A | ph } /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
5	4	abbii	|- { x | ( x e. { x e. A | ph } /\ ps ) } = { x | ( x e. A /\ ( ph /\ ps ) ) }
6		df-rab	|- { x e. { x e. A | ph } | ps } = { x | ( x e. { x e. A | ph } /\ ps ) }
7		df-rab	|- { x e. A | ( ph /\ ps ) } = { x | ( x e. A /\ ( ph /\ ps ) ) }
8	5 6 7	3eqtr4i	|- { x e. { x e. A | ph } | ps } = { x e. A | ( ph /\ ps ) }