Theorem euabsn2 4015

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 2222	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		abeq1 2507	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x e. { y } ) )
3		elsn 3958	. . . . . 6 |- ( x e. { y } <-> x = y )
4	3	bibi2i 311	. . . . 5 |- ( ( ph <-> x e. { y } ) <-> ( ph <-> x = y ) )
5	4	albii 1648	. . . 4 |- ( A. x ( ph <-> x e. { y } ) <-> A. x ( ph <-> x = y ) )
6	2, 5	bitri 249	. . 3 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
7	6	exbii 1675	. 2 |- ( E. y { x | ph } = { y } <-> E. y A. x ( ph <-> x = y ) )
8	1, 7	bitr4i 252	1 |- ( E! x ph <-> E. y { x | ph } = { y } )

Syntax hints:   <-> wb 184   A.wal 1397   = wceq 1399   E.wex 1620   e. wcel 1826   E!weu 2218   {cab 2367   {csn 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-clab 2368  df-cleq 2374  df-clel 2377  df-sn 3945

This theorem is referenced by:  euabsn  4016  reusn  4017  absneu  4018  uniintab  4238  eusvobj2  6189