Theorem euabsn2 3624

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 2006	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		abeq1 2264	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x e. { y } ) )
3		velsn 3573	. . . . . 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 1582	. 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!weu 2003   e. wcel 2125   {cab 2140   {csn 3556

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-sn 3562

This theorem is referenced by:  euabsn  3625  reusn  3626  absneu  3627  uniintabim  3840  euabex  4180  nfvres  5494  eusvobj2  5800