Theorem reusn 3678

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 3676	. 2 |- ( E! x ( x e. A /\ ph ) <-> E. y { x | ( x e. A /\ ph ) } = { y } )
2		df-reu 2475	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rab 2477	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	eqeq1i 2197	. . 3 |- ( { x e. A | ph } = { y } <-> { x | ( x e. A /\ ph ) } = { y } )
5	4	exbii 1616	. 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 1364   E.wex 1503   E!weu 2038   e. wcel 2160   {cab 2175   E!wreu 2470   {crab 2472   {csn 3607

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-reu 2475  df-rab 2477  df-v 2754  df-sn 3613

This theorem is referenced by:  reuen1  6819