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Theorem rexab2 2730
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 2329 . 2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. x ( x  e. 
{ y  |  ph }  /\  ps ) )
2 nfsab1 2046 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1437 . . . 4  |-  F/ y ps
42, 3nfan 1473 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  /\  ps )
5 nfv 1437 . . 3  |-  F/ x
( ph  /\  ch )
6 eleq1 2116 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2044 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 189 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9anbi12d 450 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  /\  ps )  <->  ( ph  /\ 
ch ) ) )
114, 5, 10cbvex 1655 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ps )  <->  E. y
( ph  /\  ch )
)
121, 11bitri 177 1  |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rex 2329
This theorem is referenced by:  rexrab2  2731
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