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Theorem reusn 3869
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 3867 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E. y { x  |  ( x  e.  A  /\  ph ) }  =  { y } )
2 df-reu 2704 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 df-rab 2706 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eqeq1i 2442 . . 3  |-  ( { x  e.  A  |  ph }  =  { y }  <->  { x  |  ( x  e.  A  /\  ph ) }  =  {
y } )
54exbii 1592 . 2  |-  ( E. y { x  e.  A  |  ph }  =  { y }  <->  E. y { x  |  (
x  e.  A  /\  ph ) }  =  {
y } )
61, 2, 53bitr4i 269 1  |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   {cab 2421   E!wreu 2699   {crab 2701   {csn 3806
This theorem is referenced by:  reusv6OLD  4726  reuen1  7168  cshwssizesame  28251  frisusgranb  28324  vdn1frgrav2  28353  vdgn1frgrav2  28354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-reu 2704  df-rab 2706  df-sn 3812
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