Theorem reusn 3647

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

Step	Hyp	Ref	Expression
1		euabsn2 3645	. 2 |- ( E! x ( x e. A /\ ph ) <-> E. y { x | ( x e. A /\ ph ) } = { y } )
2		df-reu 2451	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rab 2453	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	eqeq1i 2173	. . 3 |- ( { x e. A | ph } = { y } <-> { x | ( x e. A /\ ph ) } = { y } )
5	4	exbii 1593	. 2 |- ( E. y { x e. A | ph } = { y } <-> E. y { x | ( x e. A /\ ph ) } = { y } )
6	1, 2, 5	3bitr4i 211	1 |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   E!weu 2014   e. wcel 2136   {cab 2151   E!wreu 2446   {crab 2448   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-reu 2451  df-rab 2453  df-v 2728  df-sn 3582

This theorem is referenced by:  reuen1  6767