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Theorem rexrab2 2919
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 2477 . . 3  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
21rexeqi 2691 . 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 2918 . 2  |-  ( E. x  e.  { y  |  ( y  e.  A  /\  ph ) } ps  <->  E. y ( ( y  e.  A  /\  ph )  /\  ch )
)
5 anass 401 . . . 4  |-  ( ( ( y  e.  A  /\  ph )  /\  ch ) 
<->  ( y  e.  A  /\  ( ph  /\  ch ) ) )
65exbii 1616 . . 3  |-  ( E. y ( ( y  e.  A  /\  ph )  /\  ch )  <->  E. y
( y  e.  A  /\  ( ph  /\  ch ) ) )
7 df-rex 2474 . . 3  |-  ( E. y  e.  A  (
ph  /\  ch )  <->  E. y ( y  e.  A  /\  ( ph  /\ 
ch ) ) )
86, 7bitr4i 187 . 2  |-  ( E. y ( ( y  e.  A  /\  ph )  /\  ch )  <->  E. y  e.  A  ( ph  /\ 
ch ) )
92, 4, 83bitri 206 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 104    <-> wb 105   E.wex 1503    e. wcel 2160   {cab 2175   E.wrex 2469   {crab 2472
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-rab 2477
This theorem is referenced by: (None)
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