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Theorem rexrab 2929
Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexrab  |-  ( E. x  e.  { y  e.  A  |  ph } ch  <->  E. x  e.  A  ( ps  /\  ch )
)
Distinct variable groups:    x, y    y, A    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    ch( x, y)    A( x)

Proof of Theorem rexrab
StepHypRef Expression
1 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
21elrab 2923 . . . 4  |-  ( x  e.  { y  e.  A  |  ph }  <->  ( x  e.  A  /\  ps ) )
32anbi1i 676 . . 3  |-  ( ( x  e.  { y  e.  A  |  ph }  /\  ch )  <->  ( (
x  e.  A  /\  ps )  /\  ch )
)
4 anass 630 . . 3  |-  ( ( ( x  e.  A  /\  ps )  /\  ch ) 
<->  ( x  e.  A  /\  ( ps  /\  ch ) ) )
53, 4bitri 240 . 2  |-  ( ( x  e.  { y  e.  A  |  ph }  /\  ch )  <->  ( x  e.  A  /\  ( ps  /\  ch ) ) )
65rexbii2 2572 1  |-  ( E. x  e.  { y  e.  A  |  ph } ch  <->  E. x  e.  A  ( ps  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   E.wrex 2544   {crab 2547
This theorem is referenced by:  wereu2  4390  wdom2d  7294  enfin2i  7947  infm3  9713  pgpssslw  14925  1stcfb  17171  xkobval  17281  xkococn  17354  imasdsf1olem  17937  cvmliftlem15  23829  rexrabOLD  26356  rexrabdioph  26875  ellspd  27254  hbtlem6  27333  pmtrfrn  27400
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-rab 2552  df-v 2790
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