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Theorem euabex 4009
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 3480 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 vex 2614 . . . . 5  |-  y  e. 
_V
32snex 3978 . . . 4  |-  { y }  e.  _V
4 eleq1 2145 . . . 4  |-  ( { x  |  ph }  =  { y }  ->  ( { x  |  ph }  e.  _V  <->  { y }  e.  _V )
)
53, 4mpbiri 166 . . 3  |-  ( { x  |  ph }  =  { y }  ->  { x  |  ph }  e.  _V )
65exlimiv 1530 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  ->  { x  |  ph }  e.  _V )
71, 6sylbi 119 1  |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1943   {cab 2069   _Vcvv 2611   {csn 3417
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423
This theorem is referenced by: (None)
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