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Theorem euabex 4117
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
euabex  |-  ( E! x ph  ->  { x  |  ph }  e.  _V )

Proof of Theorem euabex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3562 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 vex 2663 . . . . 5  |-  y  e. 
_V
32snex 4079 . . . 4  |-  { y }  e.  _V
4 eleq1 2180 . . . 4  |-  ( { x  |  ph }  =  { y }  ->  ( { x  |  ph }  e.  _V  <->  { y }  e.  _V )
)
53, 4mpbiri 167 . . 3  |-  ( { x  |  ph }  =  { y }  ->  { x  |  ph }  e.  _V )
65exlimiv 1562 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  ->  { x  |  ph }  e.  _V )
71, 6sylbi 120 1  |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   E.wex 1453    e. wcel 1465   E!weu 1977   {cab 2103   _Vcvv 2660   {csn 3497
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-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503
This theorem is referenced by: (None)
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