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Theorem euabsn 4104
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 4103 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 nfv 1708 . . 3  |-  F/ y { x  |  ph }  =  { x }
3 nfab1 2621 . . . 4  |-  F/_ x { x  |  ph }
43nfeq1 2634 . . 3  |-  F/ x { x  |  ph }  =  { y }
5 sneq 4042 . . . 4  |-  ( x  =  y  ->  { x }  =  { y } )
65eqeq2d 2471 . . 3  |-  ( x  =  y  ->  ( { x  |  ph }  =  { x }  <->  { x  |  ph }  =  {
y } ) )
72, 4, 6cbvex 2023 . 2  |-  ( E. x { x  | 
ph }  =  {
x }  <->  E. y { x  |  ph }  =  { y } )
81, 7bitr4i 252 1  |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395   E.wex 1613   E!weu 2283   {cab 2442   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-sn 4033
This theorem is referenced by:  eusn  4108  uniintsn  4326  args  5375  opabiotadm  5935  mapsn  7479
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