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Theorem rabbi 2731
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2792. (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 2406 . 2  |-  ( A. x ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) )  <->  { x  |  ( x  e.  A  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ch ) } )
2 df-ral 2561 . . 3  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( x  e.  A  ->  ( ps  <->  ch )
) )
3 pm5.32 617 . . . 4  |-  ( ( x  e.  A  -> 
( ps  <->  ch )
)  <->  ( ( x  e.  A  /\  ps ) 
<->  ( x  e.  A  /\  ch ) ) )
43albii 1556 . . 3  |-  ( A. x ( x  e.  A  ->  ( ps  <->  ch ) )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
52, 4bitri 240 . 2  |-  ( A. x  e.  A  ( ps 
<->  ch )  <->  A. x
( ( x  e.  A  /\  ps )  <->  ( x  e.  A  /\  ch ) ) )
6 df-rab 2565 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
7 df-rab 2565 . . 3  |-  { x  e.  A  |  ch }  =  { x  |  ( x  e.  A  /\  ch ) }
86, 7eqeq12i 2309 . 2  |-  ( { x  e.  A  |  ps }  =  { x  e.  A  |  ch } 
<->  { x  |  ( x  e.  A  /\  ps ) }  =  {
x  |  ( x  e.  A  /\  ch ) } )
91, 5, 83bitr4i 268 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    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560
This theorem is referenced by:  rabbidva  2792  kqfeq  17431  isr0  17444  eq0rabdioph  26959  eqrabdioph  26960  lerabdioph  26989  eluzrabdioph  26990  ltrabdioph  26992  nerabdioph  26993  dvdsrabdioph  26994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-ral 2561  df-rab 2565
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