Theorem rabbi 2487

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2548. (Contributed by NM, 25-Nov-2013.)

Assertion

Ref	Expression
rabbi	|- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )

Proof of Theorem rabbi

Step	Hyp	Ref	Expression
1		abbi 2151	. 2 |- ( A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
2		df-ral 2311	. . 3 |- ( A. x e. A ( ps <-> ch ) <-> A. x ( x e. A -> ( ps <-> ch ) ) )
3		pm5.32 426	. . . 4 |- ( ( x e. A -> ( ps <-> ch ) ) <-> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	3	albii 1359	. . 3 |- ( A. x ( x e. A -> ( ps <-> ch ) ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
5	2, 4	bitri 173	. 2 |- ( A. x e. A ( ps <-> ch ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
6		df-rab 2315	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
7		df-rab 2315	. . 3 |- { x e. A | ch } = { x | ( x e. A /\ ch ) }
8	6, 7	eqeq12i 2053	. 2 |- ( { x e. A | ps } = { x e. A | ch } <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
9	1, 5, 8	3bitr4i 201	1 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-ral 2311  df-rab 2315

This theorem is referenced by:  rabbidva  2548