Theorem rabbi 2647

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2718. (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 2284	. 2 |- ( A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
2		df-ral 2453	. . 3 |- ( A. x e. A ( ps <-> ch ) <-> A. x ( x e. A -> ( ps <-> ch ) ) )
3		pm5.32 450	. . . 4 |- ( ( x e. A -> ( ps <-> ch ) ) <-> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	3	albii 1463	. . 3 |- ( A. x ( x e. A -> ( ps <-> ch ) ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
5	2, 4	bitri 183	. 2 |- ( A. x e. A ( ps <-> ch ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
6		df-rab 2457	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
7		df-rab 2457	. . 3 |- { x e. A | ch } = { x | ( x e. A /\ ch ) }
8	6, 7	eqeq12i 2184	. 2 |- ( { x e. A | ps } = { x e. A | ch } <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
9	1, 5, 8	3bitr4i 211	1 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   {crab 2452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-ral 2453  df-rab 2457

This theorem is referenced by:  rabbidva  2718  exmidonfinlem  7170