Theorem reusn 4262

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 4260	. 2 |- ( E! x ( x e. A /\ ph ) <-> E. y { x | ( x e. A /\ ph ) } = { y } )
2		df-reu 2919	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		df-rab 2921	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	3	eqeq1i 2627	. . 3 |- ( { x e. A | ph } = { y } <-> { x | ( x e. A /\ ph ) } = { y } )
5	4	exbii 1774	. 2 |- ( E. y { x e. A | ph } = { y } <-> E. y { x | ( x e. A /\ ph ) } = { y } )
6	1, 2, 5	3bitr4i 292	1 |- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   {cab 2608   E!wreu 2914   {crab 2916   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-reu 2919  df-rab 2921  df-v 3202  df-sn 4178

This theorem is referenced by:  reuen1  8025  cshwrepswhash1  15809  frcond3  27133  vdgn1frgrv2  27160  ddemeas  30299