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Theorem ralab 2845
Description: Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
ralab.1  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralab  |-  ( A. x  e.  { y  |  ph } ch  <->  A. x
( ps  ->  ch ) )
Distinct variable groups:    x, y    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    ch( x, y)

Proof of Theorem ralab
StepHypRef Expression
1 df-ral 2422 . 2  |-  ( A. x  e.  { y  |  ph } ch  <->  A. x
( x  e.  {
y  |  ph }  ->  ch ) )
2 vex 2690 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 2829 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54imbi1i 237 . . 3  |-  ( ( x  e.  { y  |  ph }  ->  ch )  <->  ( ps  ->  ch ) )
65albii 1447 . 2  |-  ( A. x ( x  e. 
{ y  |  ph }  ->  ch )  <->  A. x
( ps  ->  ch ) )
71, 6bitri 183 1  |-  ( A. x  e.  { y  |  ph } ch  <->  A. x
( ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330    e. wcel 1481   {cab 2126   A.wral 2417
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2689
This theorem is referenced by:  funcnvuni  5196  ralrnmpo  5889  pitonn  7676
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