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Theorem euabsn2 4034
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 2323 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 abeq1 2581 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  e.  { y } ) )
3 elsn 3973 . . . . . 6  |-  ( x  e.  { y }  <-> 
x  =  y )
43bibi2i 320 . . . . 5  |-  ( (
ph 
<->  x  e.  { y } )  <->  ( ph  <->  x  =  y ) )
54albii 1699 . . . 4  |-  ( A. x ( ph  <->  x  e.  { y } )  <->  A. x
( ph  <->  x  =  y
) )
62, 5bitri 257 . . 3  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
76exbii 1726 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  <->  E. y A. x ( ph  <->  x  =  y ) )
81, 7bitr4i 260 1  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   {cab 2457   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-sn 3960
This theorem is referenced by:  euabsn  4035  reusn  4036  absneu  4037  uniintab  4264  eusvobj2  6301
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