Theorem euabsn2 4034

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
euabsn2	|- ( E! x ph <-> E. y { x | ph } = { y } )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem euabsn2

Step	Hyp	Ref	Expression
1		df-eu 2323	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		abeq1 2581	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x e. { y } ) )
3		elsn 3973	. . . . . 6 |- ( x e. { y } <-> x = y )
4	3	bibi2i 320	. . . . 5 |- ( ( ph <-> x e. { y } ) <-> ( ph <-> x = y ) )
5	4	albii 1699	. . . 4 |- ( A. x ( ph <-> x e. { y } ) <-> A. x ( ph <-> x = y ) )
6	2, 5	bitri 257	. . 3 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
7	6	exbii 1726	. 2 |- ( E. y { x | ph } = { y } <-> E. y A. x ( ph <-> x = y ) )
8	1, 7	bitr4i 260	1 |- ( E! x ph <-> E. y { x | ph } = { y } )

Syntax hints:   <-> wb 189   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   E!weu 2319   {cab 2457   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-sn 3960

This theorem is referenced by:  euabsn  4035  reusn  4036  absneu  4037  uniintab  4264  eusvobj2  6301