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Theorem rabbi 2504
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2565. (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
StepHypRef Expression
1 abbi 2167 . 2  |-  ( A. x ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) )  <->  { x  |  ( x  e.  A  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ch ) } )
2 df-ral 2328 . . 3  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( x  e.  A  ->  ( ps  <->  ch )
) )
3 pm5.32 434 . . . 4  |-  ( ( x  e.  A  -> 
( ps  <->  ch )
)  <->  ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) ) )
43albii 1375 . . 3  |-  ( A. x ( x  e.  A  ->  ( ps  <->  ch ) )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
52, 4bitri 177 . 2  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
6 df-rab 2332 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
7 df-rab 2332 . . 3  |-  { x  e.  A  |  ch }  =  { x  |  ( x  e.  A  /\  ch ) }
86, 7eqeq12i 2069 . 2  |-  ( { x  e.  A  |  ps }  =  { x  e.  A  |  ch } 
<->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  A  /\  ch ) } )
91, 5, 83bitr4i 205 1  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  { x  e.  A  |  ps }  =  { x  e.  A  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-ral 2328  df-rab 2332
This theorem is referenced by:  rabbidva  2565
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