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Theorem ralab 2765
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 2360 . 2  |-  ( A. x  e.  { y  |  ph } ch  <->  A. x
( x  e.  {
y  |  ph }  ->  ch ) )
2 vex 2617 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 2750 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54imbi1i 236 . . 3  |-  ( ( x  e.  { y  |  ph }  ->  ch )  <->  ( ps  ->  ch ) )
65albii 1402 . 2  |-  ( A. x ( x  e. 
{ y  |  ph }  ->  ch )  <->  A. x
( ps  ->  ch ) )
71, 6bitri 182 1  |-  ( A. x  e.  { y  |  ph } ch  <->  A. x
( ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1285    e. wcel 1436   {cab 2071   A.wral 2355
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616
This theorem is referenced by:  funcnvuni  5039  ralrnmpt2  5697  pitonn  7306
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