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Theorem euabsn 3900
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 3899 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 nfv 1630 . . 3  |-  F/ y { x  |  ph }  =  { x }
3 nfab1 2580 . . . 4  |-  F/_ x { x  |  ph }
43nfeq1 2587 . . 3  |-  F/ x { x  |  ph }  =  { y }
5 sneq 3849 . . . 4  |-  ( x  =  y  ->  { x }  =  { y } )
65eqeq2d 2453 . . 3  |-  ( x  =  y  ->  ( { x  |  ph }  =  { x }  <->  { x  |  ph }  =  {
y } ) )
72, 4, 6cbvex 1986 . 2  |-  ( E. x { x  | 
ph }  =  {
x }  <->  E. y { x  |  ph }  =  { y } )
81, 7bitr4i 245 1  |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653   E!weu 2287   {cab 2428   {csn 3838
This theorem is referenced by:  eusn  3904  uniintsn  4111  args  5261  opabiotadm  6566  mapsn  7084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-sn 3844
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