MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusn Unicode version

Theorem reusn 3820
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 3818 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E. y { x  |  ( x  e.  A  /\  ph ) }  =  { y } )
2 df-reu 2656 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 df-rab 2658 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43eqeq1i 2394 . . 3  |-  ( { x  e.  A  |  ph }  =  { y }  <->  { x  |  ( x  e.  A  /\  ph ) }  =  {
y } )
54exbii 1589 . 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 1547    = wceq 1649    e. wcel 1717   E!weu 2238   {cab 2373   E!wreu 2651   {crab 2653   {csn 3757
This theorem is referenced by:  reusv6OLD  4674  reuen1  7112  frisusgranb  27750  vdn1frgrav2  27779  vdgn1frgrav2  27780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-reu 2656  df-rab 2658  df-sn 3763
  Copyright terms: Public domain W3C validator