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Theorem rabbi 2544
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2607. (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 2201 . 2  |-  ( A. x ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) )  <->  { x  |  ( x  e.  A  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ch ) } )
2 df-ral 2364 . . 3  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( x  e.  A  ->  ( ps  <->  ch )
) )
3 pm5.32 441 . . . 4  |-  ( ( x  e.  A  -> 
( ps  <->  ch )
)  <->  ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) ) )
43albii 1404 . . 3  |-  ( A. x ( x  e.  A  ->  ( ps  <->  ch ) )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
52, 4bitri 182 . 2  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
6 df-rab 2368 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
7 df-rab 2368 . . 3  |-  { x  e.  A  |  ch }  =  { x  |  ( x  e.  A  /\  ch ) }
86, 7eqeq12i 2101 . 2  |-  ( { x  e.  A  |  ps }  =  { x  e.  A  |  ch } 
<->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  A  /\  ch ) } )
91, 5, 83bitr4i 210 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 102    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   {crab 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-ral 2364  df-rab 2368
This theorem is referenced by:  rabbidva  2607
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