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Theorem ralrab 2762
Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
ralab.1  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralrab  |-  ( A. x  e.  { y  e.  A  |  ph } ch 
<-> 
A. x  e.  A  ( ps  ->  ch )
)
Distinct variable groups:    x, y    y, A    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    ch( x, y)    A( x)

Proof of Theorem ralrab
StepHypRef Expression
1 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
21elrab 2757 . . . 4  |-  ( x  e.  { y  e.  A  |  ph }  <->  ( x  e.  A  /\  ps ) )
32imbi1i 236 . . 3  |-  ( ( x  e.  { y  e.  A  |  ph }  ->  ch )  <->  ( (
x  e.  A  /\  ps )  ->  ch )
)
4 impexp 259 . . 3  |-  ( ( ( x  e.  A  /\  ps )  ->  ch ) 
<->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
53, 4bitri 182 . 2  |-  ( ( x  e.  { y  e.  A  |  ph }  ->  ch )  <->  ( x  e.  A  ->  ( ps 
->  ch ) ) )
65ralbii2 2381 1  |-  ( A. x  e.  { y  e.  A  |  ph } ch 
<-> 
A. x  e.  A  ( ps  ->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   A.wral 2353   {crab 2357
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2612
This theorem is referenced by: (None)
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