Theorem rabbi 2672

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2748. (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 2307	. 2 |- ( A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
2		df-ral 2477	. . 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 1481	. . 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 2481	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
7		df-rab 2481	. . 3 |- { x e. A | ch } = { x | ( x e. A /\ ch ) }
8	6, 7	eqeq12i 2207	. 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 1362   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   {crab 2476

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-ral 2477  df-rab 2481

This theorem is referenced by:  rabbidva  2748  exmidonfinlem  7253