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Theorem rexab2 2803
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 2381 . 2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. x ( x  e. 
{ y  |  ph }  /\  ps ) )
2 nfsab1 2090 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1476 . . . 4  |-  F/ y ps
42, 3nfan 1512 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  /\  ps )
5 nfv 1476 . . 3  |-  F/ x
( ph  /\  ch )
6 eleq1 2162 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2088 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 195 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9anbi12d 460 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  /\  ps )  <->  ( ph  /\ 
ch ) ) )
114, 5, 10cbvex 1697 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ps )  <->  E. y
( ph  /\  ch )
)
121, 11bitri 183 1  |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1436    e. wcel 1448   {cab 2086   E.wrex 2376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-rex 2381
This theorem is referenced by:  rexrab2  2804
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