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Theorem absneu 3714
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 3664 . . . . 5  |-  ( y  =  A  ->  { y }  =  { A } )
21eqeq2d 2307 . . . 4  |-  ( y  =  A  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  { A } ) )
32spcegv 2882 . . 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 3711 . 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 1531    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   {csn 3653
This theorem is referenced by:  rabsneu  3715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sn 3659
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