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Theorem euabsn 3699
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 3698 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 nfv 1605 . . 3  |-  F/ y { x  |  ph }  =  { x }
3 nfab1 2421 . . . 4  |-  F/_ x { x  |  ph }
43nfeq1 2428 . . 3  |-  F/ x { x  |  ph }  =  { y }
5 sneq 3651 . . . 4  |-  ( x  =  y  ->  { x }  =  { y } )
65eqeq2d 2294 . . 3  |-  ( x  =  y  ->  ( { x  |  ph }  =  { x }  <->  { x  |  ph }  =  {
y } ) )
72, 4, 6cbvex 1925 . 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 1528    = wceq 1623   E!weu 2143   {cab 2269   {csn 3640
This theorem is referenced by:  eusn  3703  uniintsn  3899  args  5041  opabiotadm  6292  mapsn  6809
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-or 359  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-nfc 2408  df-sn 3646
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