Theorem rabrab 29338

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

Assertion

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

Proof of Theorem rabrab

Step	Hyp	Ref	Expression
1		rabid 3116	. . . . 5 |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) )
2	1	anbi1i 731	. . . 4 |- ( ( x e. { x e. A | ph } /\ ps ) <-> ( ( x e. A /\ ph ) /\ ps ) )
3		anass 681	. . . 4 |- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
4	2, 3	bitri 264	. . 3 |- ( ( x e. { x e. A | ph } /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
5	4	abbii 2739	. 2 |- { x | ( x e. { x e. A | ph } /\ ps ) } = { x | ( x e. A /\ ( ph /\ ps ) ) }
6		df-rab 2921	. 2 |- { x e. { x e. A | ph } | ps } = { x | ( x e. { x e. A | ph } /\ ps ) }
7		df-rab 2921	. 2 |- { x e. A | ( ph /\ ps ) } = { x | ( x e. A /\ ( ph /\ ps ) ) }
8	5, 6, 7	3eqtr4i 2654	1 |- { x e. { x e. A | ph } | ps } = { x e. A | ( ph /\ ps ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921

This theorem is referenced by:  fpwrelmapffs  29509