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Theorem rexab 3233
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 2777 . 2  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( x  e. 
{ y  |  ph }  /\  ch ) )
2 vex 3083 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 3217 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54anbi1i 699 . . 3  |-  ( ( x  e.  { y  |  ph }  /\  ch )  <->  ( ps  /\  ch ) )
65exbii 1712 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ch )  <->  E. x
( ps  /\  ch ) )
71, 6bitri 252 1  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   E.wex 1657    e. wcel 1872   {cab 2407   E.wrex 2772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-v 3082
This theorem is referenced by:  4sqlem12  14899  nofulllem5  30600  mblfinlem3  31943  mblfinlem4  31944  ismblfin  31945  itg2addnclem  31957  itg2addnc  31960  diophrex  35587
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