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Theorem rexab 2928
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 2549 . 2  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( x  e. 
{ y  |  ph }  /\  ch ) )
2 vex 2791 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 2914 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54anbi1i 676 . . 3  |-  ( ( x  e.  { y  |  ph }  /\  ch )  <->  ( ps  /\  ch ) )
65exbii 1569 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ch )  <->  E. x
( ps  /\  ch ) )
71, 6bitri 240 1  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   {cab 2269   E.wrex 2544
This theorem is referenced by:  4sqlem12  13003  nofulllem5  24360  diophrex  26855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790
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