Theorem rabbi 2655

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2727. (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 2291	. 2 |- ( A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
2		df-ral 2460	. . 3 |- ( A. x e. A ( ps <-> ch ) <-> A. x ( x e. A -> ( ps <-> ch ) ) )
3		pm5.32 453	. . . 4 |- ( ( x e. A -> ( ps <-> ch ) ) <-> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	3	albii 1470	. . 3 |- ( A. x ( x e. A -> ( ps <-> ch ) ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
5	2, 4	bitri 184	. 2 |- ( A. x e. A ( ps <-> ch ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
6		df-rab 2464	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
7		df-rab 2464	. . 3 |- { x e. A | ch } = { x | ( x e. A /\ ch ) }
8	6, 7	eqeq12i 2191	. 2 |- ( { x e. A | ps } = { x e. A | ch } <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
9	1, 5, 8	3bitr4i 212	1 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   {crab 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-ral 2460  df-rab 2464

This theorem is referenced by:  rabbidva  2727  exmidonfinlem  7194