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Theorem absneu 3590
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 3533 . . . . 5  |-  ( y  =  A  ->  { y }  =  { A } )
21eqeq2d 2149 . . . 4  |-  ( y  =  A  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  { A } ) )
32spcegv 2769 . . 3  |-  ( A  e.  V  ->  ( { x  |  ph }  =  { A }  ->  E. y { x  | 
ph }  =  {
y } ) )
43imp 123 . 2  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E. y { x  |  ph }  =  {
y } )
5 euabsn2 3587 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
64, 5sylibr 133 1  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1997   {cab 2123   {csn 3522
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sn 3528
This theorem is referenced by:  rabsneu  3591
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