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Theorem reusn 3713
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 3711 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E. y { x  |  ( x  e.  A  /\  ph ) }  =  { y } )
2 df-reu 2563 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 df-rab 2565 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eqeq1i 2303 . . 3  |-  ( { x  e.  A  |  ph }  =  { y }  <->  { x  |  ( x  e.  A  /\  ph ) }  =  {
y } )
54exbii 1572 . 2  |-  ( E. y { x  e.  A  |  ph }  =  { y }  <->  E. y { x  |  (
x  e.  A  /\  ph ) }  =  {
y } )
61, 2, 53bitr4i 268 1  |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   E!wreu 2558   {crab 2560   {csn 3653
This theorem is referenced by:  reusv6OLD  4561  reuen1  6946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-reu 2563  df-rab 2565  df-sn 3659
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