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Metamath Proof Explorer

Theorem rabbi

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

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

Proof

Step Hyp Ref Expression
1 abbib 
 |-  ( { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
2 df-rab 
 |-  { x e. A | ps } = { x | ( x e. A /\ ps ) }
3 df-rab 
 |-  { x e. A | ch } = { x | ( x e. A /\ ch ) }
4 2 3 eqeq12i 
 |-  ( { x e. A | ps } = { x e. A | ch } <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
5 df-ral 
 |-  ( A. x e. A ( ps <-> ch ) <-> A. x ( x e. A -> ( ps <-> ch ) ) )
6 pm5.32 
 |-  ( ( x e. A -> ( ps <-> ch ) ) <-> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
7 6 albii 
 |-  ( A. x ( x e. A -> ( ps <-> ch ) ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
8 5 7 bitri 
 |-  ( A. x e. A ( ps <-> ch ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
9 1 4 8 3bitr4ri 
 |-  ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )