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Theorem rexab2 3266
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexab2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\  ch ) )
Distinct variable groups:    x, y    ch, x    ph, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)    ch( y)

Proof of Theorem rexab2
StepHypRef Expression
1 df-rex 2813 . 2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. x ( x  e. 
{ y  |  ph }  /\  ps ) )
2 nfsab1 2446 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1708 . . . 4  |-  F/ y ps
42, 3nfan 1929 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  /\  ps )
5 nfv 1708 . . 3  |-  F/ x
( ph  /\  ch )
6 eleq1 2529 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2444 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 261 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9anbi12d 710 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  /\  ps )  <->  ( ph  /\ 
ch ) ) )
114, 5, 10cbvex 2023 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ps )  <->  E. y
( ph  /\  ch )
)
121, 11bitri 249 1  |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1613    e. wcel 1819   {cab 2442   E.wrex 2808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-rex 2813
This theorem is referenced by:  rexrab2  3267  tmdgsum2  20721
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