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Theorem rabbi 2583
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2646. (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 2229 . 2  |-  ( A. x ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) )  <->  { x  |  ( x  e.  A  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ch ) } )
2 df-ral 2396 . . 3  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( x  e.  A  ->  ( ps  <->  ch )
) )
3 pm5.32 446 . . . 4  |-  ( ( x  e.  A  -> 
( ps  <->  ch )
)  <->  ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) ) )
43albii 1429 . . 3  |-  ( A. x ( x  e.  A  ->  ( ps  <->  ch ) )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
52, 4bitri 183 . 2  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
6 df-rab 2400 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
7 df-rab 2400 . . 3  |-  { x  e.  A  |  ch }  =  { x  |  ( x  e.  A  /\  ch ) }
86, 7eqeq12i 2129 . 2  |-  ( { x  e.  A  |  ps }  =  { x  e.  A  |  ch } 
<->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  A  /\  ch ) } )
91, 5, 83bitr4i 211 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 103    <-> wb 104   A.wal 1312    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   {crab 2395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-ral 2396  df-rab 2400
This theorem is referenced by:  rabbidva  2646
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