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Theorem rexrab2 2780
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
rexrab2  |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. y  e.  A  ( ph  /\  ch )
)
Distinct variable groups:    x, y    x, A    ch, x    ph, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)    ch( y)    A( y)

Proof of Theorem rexrab2
StepHypRef Expression
1 df-rab 2368 . . 3  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
21rexeqi 2567 . 2  |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. x  e.  {
y  |  ( y  e.  A  /\  ph ) } ps )
3 ralab2.1 . . 3  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
43rexab2 2779 . 2  |-  ( E. x  e.  { y  |  ( y  e.  A  /\  ph ) } ps  <->  E. y ( ( y  e.  A  /\  ph )  /\  ch )
)
5 anass 393 . . . 4  |-  ( ( ( y  e.  A  /\  ph )  /\  ch ) 
<->  ( y  e.  A  /\  ( ph  /\  ch ) ) )
65exbii 1541 . . 3  |-  ( E. y ( ( y  e.  A  /\  ph )  /\  ch )  <->  E. y
( y  e.  A  /\  ( ph  /\  ch ) ) )
7 df-rex 2365 . . 3  |-  ( E. y  e.  A  (
ph  /\  ch )  <->  E. y ( y  e.  A  /\  ( ph  /\ 
ch ) ) )
86, 7bitr4i 185 . 2  |-  ( E. y ( ( y  e.  A  /\  ph )  /\  ch )  <->  E. y  e.  A  ( ph  /\ 
ch ) )
92, 4, 83bitri 204 1  |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. y  e.  A  ( ph  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360   {crab 2363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-rab 2368
This theorem is referenced by: (None)
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