Theorem reusn 3737

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 3735	. 2 |- ( E! x ( x e. A /\ ph ) <-> E. y { x | ( x e. A /\ ph ) } = { y } )
2		df-reu 2515	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rab 2517	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	eqeq1i 2237	. . 3 |- ( { x e. A | ph } = { y } <-> { x | ( x e. A /\ ph ) } = { y } )
5	4	exbii 1651	. 2 |- ( E. y { x e. A | ph } = { y } <-> E. y { x | ( x e. A /\ ph ) } = { y } )
6	1, 2, 5	3bitr4i 212	1 |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   {cab 2215   E!wreu 2510   {crab 2512   {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-reu 2515  df-rab 2517  df-v 2801  df-sn 3672

This theorem is referenced by:  reuen1  6953