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Theorem euabsn 3651
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 3650 . 2  |-  ( E! x ph  <->  E. y { x  |  ph }  =  { y } )
2 nfv 1521 . . 3  |-  F/ y { x  |  ph }  =  { x }
3 nfab1 2314 . . . 4  |-  F/_ x { x  |  ph }
43nfeq1 2322 . . 3  |-  F/ x { x  |  ph }  =  { y }
5 sneq 3592 . . . 4  |-  ( x  =  y  ->  { x }  =  { y } )
65eqeq2d 2182 . . 3  |-  ( x  =  y  ->  ( { x  |  ph }  =  { x }  <->  { x  |  ph }  =  {
y } ) )
72, 4, 6cbvex 1749 . 2  |-  ( E. x { x  | 
ph }  =  {
x }  <->  E. y { x  |  ph }  =  { y } )
81, 7bitr4i 186 1  |-  ( E! x ph  <->  E. x { x  |  ph }  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1348   E.wex 1485   E!weu 2019   {cab 2156   {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sn 3587
This theorem is referenced by:  eusn  3655  args  4978  mapsn  6666
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