Theorem euabsn2 3439

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 1903	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		abeq1 2147	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x e. { y } ) )
3		velsn 3392	. . . . . 6 |- ( x e. { y } <-> x = y )
4	3	bibi2i 216	. . . . 5 |- ( ( ph <-> x e. { y } ) <-> ( ph <-> x = y ) )
5	4	albii 1359	. . . 4 |- ( A. x ( ph <-> x e. { y } ) <-> A. x ( ph <-> x = y ) )
6	2, 5	bitri 173	. . 3 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
7	6	exbii 1496	. 2 |- ( E. y { x | ph } = { y } <-> E. y A. x ( ph <-> x = y ) )
8	1, 7	bitr4i 176	1 |- ( E! x ph <-> E. y { x | ph } = { y } )

Syntax hints:   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   E!weu 1900   {cab 2026   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  euabsn  3440  reusn  3441  absneu  3442  uniintabim  3652  euabex  3961  nfvres  5206  eusvobj2  5498