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Theorem absneu 3701
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
Assertion
Ref Expression
absneu  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E! x ph )

Proof of Theorem absneu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sneq 3651 . . . . 5  |-  ( y  =  A  ->  { y }  =  { A } )
21eqeq2d 2294 . . . 4  |-  ( y  =  A  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  { A } ) )
32spcegv 2869 . . 3  |-  ( A  e.  V  ->  ( { x  |  ph }  =  { A }  ->  E. y { x  | 
ph }  =  {
y } ) )
43imp 418 . 2  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E. y { x  |  ph }  =  {
y } )
5 euabsn2 3698 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
64, 5sylibr 203 1  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   {csn 3640
This theorem is referenced by:  rabsneu  3702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sn 3646
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