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Theorem rexab 2819
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 2399 . 2  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( x  e. 
{ y  |  ph }  /\  ch ) )
2 vex 2663 . . . . 5  |-  x  e. 
_V
3 ralab.1 . . . . 5  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
42, 3elab 2802 . . . 4  |-  ( x  e.  { y  | 
ph }  <->  ps )
54anbi1i 453 . . 3  |-  ( ( x  e.  { y  |  ph }  /\  ch )  <->  ( ps  /\  ch ) )
65exbii 1569 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ch )  <->  E. x
( ps  /\  ch ) )
71, 6bitri 183 1  |-  ( E. x  e.  { y  |  ph } ch  <->  E. x ( ps  /\  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1453    e. wcel 1465   {cab 2103   E.wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662
This theorem is referenced by:  rexrnmpo  5854
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