Theorem euabsn2 3600

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 2003	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		abeq1 2250	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x e. { y } ) )
3		velsn 3549	. . . . . 6 |- ( x e. { y } <-> x = y )
4	3	bibi2i 226	. . . . 5 |- ( ( ph <-> x e. { y } ) <-> ( ph <-> x = y ) )
5	4	albii 1447	. . . 4 |- ( A. x ( ph <-> x e. { y } ) <-> A. x ( ph <-> x = y ) )
6	2, 5	bitri 183	. . 3 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
7	6	exbii 1585	. 2 |- ( E. y { x | ph } = { y } <-> E. y A. x ( ph <-> x = y ) )
8	1, 7	bitr4i 186	1 |- ( E! x ph <-> E. y { x | ph } = { y } )

Syntax hints:   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   E!weu 2000   {cab 2126   {csn 3532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538

This theorem is referenced by:  euabsn  3601  reusn  3602  absneu  3603  uniintabim  3816  euabex  4155  nfvres  5462  eusvobj2  5768