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Theorem absneu 3442
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 3386 . . . . 5  |-  ( y  =  A  ->  { y }  =  { A } )
21eqeq2d 2051 . . . 4  |-  ( y  =  A  ->  ( { x  |  ph }  =  { y }  <->  { x  |  ph }  =  { A } ) )
32spcegv 2641 . . 3  |-  ( A  e.  V  ->  ( { x  |  ph }  =  { A }  ->  E. y { x  | 
ph }  =  {
y } ) )
43imp 115 . 2  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E. y { x  |  ph }  =  {
y } )
5 euabsn2 3439 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
64, 5sylibr 137 1  |-  ( ( A  e.  V  /\  { x  |  ph }  =  { A } )  ->  E! x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   E!weu 1900   {cab 2026   {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381
This theorem is referenced by:  rabsneu  3443
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