Theorem euabsn2 3792

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!xph <-> 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 2208	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
2		abeq1 2460	. . . 4 |- ({x | ph} = {y} <-> A.x(ph <-> x e. {y}))
3		elsn 3749	. . . . . 6 |- (x e. {y} <-> x = y)
4	3	bibi2i 304	. . . . 5 |- ((ph <-> x e. {y}) <-> (ph <-> x = y))
5	4	albii 1566	. . . 4 |- (A.x(ph <-> x e. {y}) <-> A.x(ph <-> x = y))
6	2, 5	bitri 240	. . 3 |- ({x | ph} = {y} <-> A.x(ph <-> x = y))
7	6	exbii 1582	. 2 |- (E.y{x | ph} = {y} <-> E.yA.x(ph <-> x = y))
8	1, 7	bitr4i 243	1 |- (E!xph <-> E.y{x | ph} = {y})

Syntax hints:   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  {cab 2339  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-sn 3742

This theorem is referenced by:  euabsn  3793  reusn  3794  absneu  3795  uniintab  3965