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Theorem rexab2 2918
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 2474 . 2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. x ( x  e. 
{ y  |  ph }  /\  ps ) )
2 nfsab1 2179 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1539 . . . 4  |-  F/ y ps
42, 3nfan 1576 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  /\  ps )
5 nfv 1539 . . 3  |-  F/ x
( ph  /\  ch )
6 eleq1 2252 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2177 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7bitrdi 196 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9anbi12d 473 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  /\  ps )  <->  ( ph  /\ 
ch ) ) )
114, 5, 10cbvex 1767 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ps )  <->  E. y
( ph  /\  ch )
)
121, 11bitri 184 1  |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   E.wex 1503    e. wcel 2160   {cab 2175   E.wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-rex 2474
This theorem is referenced by:  rexrab2  2919
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