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Theorem rexab 2775
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexab  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  ch ) )
Distinct variable groups:    x, y    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    ch( x, y)

Proof of Theorem rexab
StepHypRef Expression
1 df-rex 2365 . 2  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( x  e. 
{ y  |  ph }  /\  ch ) )
2 vex 2622 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 2758 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54anbi1i 446 . . 3  |-  ( ( x  e.  { y  |  ph }  /\  ch )  <->  ( ps  /\  ch ) )
65exbii 1541 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ch )  <->  E. x
( ps  /\  ch ) )
71, 6bitri 182 1  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  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
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-v 2621
This theorem is referenced by:  rexrnmpt2  5742
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