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Theorem absneu 3643
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 3582 . . . . 5  |-  ( y  =  A  ->  { y }  =  { A } )
21eqeq2d 2176 . . . 4  |-  ( y  =  A  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  { A } ) )
32spcegv 2810 . . 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 3640 . 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 1342   E.wex 1479   E!weu 2013    e. wcel 2135   {cab 2150   {csn 3571
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-sn 3577
This theorem is referenced by:  rabsneu  3644
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