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Theorem euabsn 3775
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 3774 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 nfv 1619 . . 3  |-  F/ y { x  |  ph }  =  { x }
3 nfab1 2496 . . . 4  |-  F/_ x { x  |  ph }
43nfeq1 2503 . . 3  |-  F/ x { x  |  ph }  =  { y }
5 sneq 3727 . . . 4  |-  ( x  =  y  ->  { x }  =  { y } )
65eqeq2d 2369 . . 3  |-  ( x  =  y  ->  ( { x  |  ph }  =  { x }  <->  { x  |  ph }  =  {
y } ) )
72, 4, 6cbvex 1990 . 2  |-  ( E. x { x  | 
ph }  =  {
x }  <->  E. y { x  |  ph }  =  { y } )
81, 7bitr4i 243 1  |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1541    = wceq 1642   E!weu 2209   {cab 2344   {csn 3716
This theorem is referenced by:  eusn  3779  uniintsn  3978  args  5120  opabiotadm  6376  mapsn  6894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-sn 3722
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