Theorem reusn 3441

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 3439	. 2 |- ( E! x ( x e. A /\ ph ) <-> E. y { x | ( x e. A /\ ph ) } = { y } )
2		df-reu 2313	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rab 2315	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	eqeq1i 2047	. . 3 |- ( { x e. A | ph } = { y } <-> { x | ( x e. A /\ ph ) } = { y } )
5	4	exbii 1496	. 2 |- ( E. y { x e. A | ph } = { y } <-> E. y { x | ( x e. A /\ ph ) } = { y } )
6	1, 2, 5	3bitr4i 201	1 |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   {cab 2026   E!wreu 2308   {crab 2310   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-reu 2313  df-rab 2315  df-v 2559  df-sn 3381

This theorem is referenced by:  reuen1  6281