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Theorem euabex 4222
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 3660 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 vex 2740 . . . . 5  |-  y  e. 
_V
32snex 4182 . . . 4  |-  { y }  e.  _V
4 eleq1 2240 . . . 4  |-  ( { x  |  ph }  =  { y }  ->  ( { x  |  ph }  e.  _V  <->  { y }  e.  _V )
)
53, 4mpbiri 168 . . 3  |-  ( { x  |  ph }  =  { y }  ->  { x  |  ph }  e.  _V )
65exlimiv 1598 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  ->  { x  |  ph }  e.  _V )
71, 6sylbi 121 1  |-  ( E! x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   E.wex 1492   E!weu 2026    e. wcel 2148   {cab 2163   _Vcvv 2737   {csn 3591
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597
This theorem is referenced by: (None)
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