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Theorem rexab 3103
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexab  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  ch ) )
Distinct variable groups:    x, y    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    ch( x, y)

Proof of Theorem rexab
StepHypRef Expression
1 df-rex 2717 . 2  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( x  e. 
{ y  |  ph }  /\  ch ) )
2 vex 2965 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 3088 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54anbi1i 678 . . 3  |-  ( ( x  e.  { y  |  ph }  /\  ch )  <->  ( ps  /\  ch ) )
65exbii 1593 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ch )  <->  E. x
( ps  /\  ch ) )
71, 6bitri 242 1  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    e. wcel 1727   {cab 2428   E.wrex 2712
This theorem is referenced by:  4sqlem12  13355  nofulllem5  25692  mblfinlem3  26281  mblfinlem4  26282  ismblfin  26283  itg2addnclem  26294  itg2addnc  26297  diophrex  26872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2717  df-v 2964
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