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Theorem euabsn2 3479
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
euabsn2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem euabsn2
StepHypRef Expression
1 df-eu 1946 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 abeq1 2192 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  e.  { y } ) )
3 velsn 3433 . . . . . 6  |-  ( x  e.  { y }  <-> 
x  =  y )
43bibi2i 225 . . . . 5  |-  ( (
ph 
<->  x  e.  { y } )  <->  ( ph  <->  x  =  y ) )
54albii 1400 . . . 4  |-  ( A. x ( ph  <->  x  e.  { y } )  <->  A. x
( ph  <->  x  =  y
) )
62, 5bitri 182 . . 3  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
76exbii 1537 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  <->  E. y A. x ( ph  <->  x  =  y ) )
81, 7bitr4i 185 1  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1943   {cab 2069   {csn 3416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sn 3422
This theorem is referenced by:  euabsn  3480  reusn  3481  absneu  3482  uniintabim  3693  euabex  4008  nfvres  5258  eusvobj2  5549
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