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Theorem euabsn2 3867
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 2284 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 abeq1 2541 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  e.  { y } ) )
3 elsn 3821 . . . . . 6  |-  ( x  e.  { y }  <-> 
x  =  y )
43bibi2i 305 . . . . 5  |-  ( (
ph 
<->  x  e.  { y } )  <->  ( ph  <->  x  =  y ) )
54albii 1575 . . . 4  |-  ( A. x ( ph  <->  x  e.  { y } )  <->  A. x
( ph  <->  x  =  y
) )
62, 5bitri 241 . . 3  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
76exbii 1592 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  <->  E. y A. x ( ph  <->  x  =  y ) )
81, 7bitr4i 244 1  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   {cab 2421   {csn 3806
This theorem is referenced by:  euabsn  3868  reusn  3869  absneu  3870  uniintab  4080  eusvobj2  6574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-sn 3812
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