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Theorem reusn 3700
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 3698 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E. y { x  |  ( x  e.  A  /\  ph ) }  =  { y } )
2 df-reu 2550 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 df-rab 2552 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eqeq1i 2290 . . 3  |-  ( { x  e.  A  |  ph }  =  { y }  <->  { x  |  ( x  e.  A  /\  ph ) }  =  {
y } )
54exbii 1569 . 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 1528    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   E!wreu 2545   {crab 2547   {csn 3640
This theorem is referenced by:  reusv6OLD  4545  reuen1  6930
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-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-reu 2550  df-rab 2552  df-sn 3646
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