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Theorem reusn 3481
Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
reusn  |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
Distinct variable groups:    x, y    ph, y    y, A
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem reusn
StepHypRef Expression
1 euabsn2 3479 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E. y { x  |  ( x  e.  A  /\  ph ) }  =  { y } )
2 df-reu 2360 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 df-rab 2362 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eqeq1i 2090 . . 3  |-  ( { x  e.  A  |  ph }  =  { y }  <->  { x  |  ( x  e.  A  /\  ph ) }  =  {
y } )
54exbii 1537 . 2  |-  ( E. y { x  e.  A  |  ph }  =  { y }  <->  E. y { x  |  (
x  e.  A  /\  ph ) }  =  {
y } )
61, 2, 53bitr4i 210 1  |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1943   {cab 2069   E!wreu 2355   {crab 2357   {csn 3416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-reu 2360  df-rab 2362  df-v 2612  df-sn 3422
This theorem is referenced by:  reuen1  6370
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