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Theorem euabsn2 3698
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 2147 . 2  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
2 abeq1 2389 . . . 4  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  e.  { y } ) )
3 elsn 3655 . . . . . 6  |-  ( x  e.  { y }  <-> 
x  =  y )
43bibi2i 304 . . . . 5  |-  ( (
ph 
<->  x  e.  { y } )  <->  ( ph  <->  x  =  y ) )
54albii 1553 . . . 4  |-  ( A. x ( ph  <->  x  e.  { y } )  <->  A. x
( ph  <->  x  =  y
) )
62, 5bitri 240 . . 3  |-  ( { x  |  ph }  =  { y }  <->  A. x
( ph  <->  x  =  y
) )
76exbii 1569 . 2  |-  ( E. y { x  | 
ph }  =  {
y }  <->  E. y A. x ( ph  <->  x  =  y ) )
81, 7bitr4i 243 1  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   {csn 3640
This theorem is referenced by:  euabsn  3699  reusn  3700  absneu  3701  uniintab  3900  eusvobj2  6337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-sn 3646
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