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Theorem euabsn 3844
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
Assertion
Ref Expression
euabsn  |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x } )

Proof of Theorem euabsn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 euabsn2 3843 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 nfv 1626 . . 3  |-  F/ y { x  |  ph }  =  { x }
3 nfab1 2550 . . . 4  |-  F/_ x { x  |  ph }
43nfeq1 2557 . . 3  |-  F/ x { x  |  ph }  =  { y }
5 sneq 3793 . . . 4  |-  ( x  =  y  ->  { x }  =  { y } )
65eqeq2d 2423 . . 3  |-  ( x  =  y  ->  ( { x  |  ph }  =  { x }  <->  { x  |  ph }  =  {
y } ) )
72, 4, 6cbvex 2060 . 2  |-  ( E. x { x  | 
ph }  =  {
x }  <->  E. y { x  |  ph }  =  { y } )
81, 7bitr4i 244 1  |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    = wceq 1649   E!weu 2262   {cab 2398   {csn 3782
This theorem is referenced by:  eusn  3848  uniintsn  4055  args  5199  opabiotadm  6504  mapsn  7022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-sn 3788
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