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Theorem euabsn2 4015
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 2222 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 abeq1 2507 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  e.  { y } ) )
3 elsn 3958 . . . . . 6  |-  ( x  e.  { y }  <-> 
x  =  y )
43bibi2i 311 . . . . 5  |-  ( (
ph 
<->  x  e.  { y } )  <->  ( ph  <->  x  =  y ) )
54albii 1648 . . . 4  |-  ( A. x ( ph  <->  x  e.  { y } )  <->  A. x
( ph  <->  x  =  y
) )
62, 5bitri 249 . . 3  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
76exbii 1675 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  <->  E. y A. x ( ph  <->  x  =  y ) )
81, 7bitr4i 252 1  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1397    = wceq 1399   E.wex 1620    e. wcel 1826   E!weu 2218   {cab 2367   {csn 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-clab 2368  df-cleq 2374  df-clel 2377  df-sn 3945
This theorem is referenced by:  euabsn  4016  reusn  4017  absneu  4018  uniintab  4238  eusvobj2  6189
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