Theorem eusn 1913

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton.

Assertion

Ref	Expression
eusn	|- (E!xph <-> E.x{x | ph} = {x})

Proof of Theorem eusn

Step	Hyp	Ref	Expression
1		cleqabr 1175	. . . 4 |- ({x | ph} = {y} <-> A.x(ph <-> x e. {y}))
2		elsn 1820	. . . . . 6 |- (x e. {y} <-> x = y)
3	2	bibi2i 460	. . . . 5 |- ((ph <-> x e. {y}) <-> (ph <-> x = y))
4	3	bial 695	. . . 4 |- (A.x(ph <-> x e. {y}) <-> A.x(ph <-> x = y))
5	1, 4	bitr 151	. . 3 |- ({x | ph} = {y} <-> A.x(ph <-> x = y))
6	5	biex 733	. 2 |- (E.y{x | ph} = {y} <-> E.yA.x(ph <-> x = y))
7		ax-17 925	. . 3 |- ({x | ph} = {x} -> A.y{x | ph} = {x})
8		hbab1 1095	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
9		ax-17 925	. . . 4 |- (z e. {y} -> A.x z e. {y})
10	8, 9	hbeq 1171	. . 3 |- ({x | ph} = {y} -> A.x{x | ph} = {y})
11		sneq 1816	. . . 4 |- (x = y -> {x} = {y})
12	11	cleq2d 1112	. . 3 |- (x = y -> ({x | ph} = {x} <-> {x | ph} = {y}))
13	7, 10, 12	cbvex 849	. 2 |- (E.x{x | ph} = {x} <-> E.y{x | ph} = {y})
14		df-eu 1009	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
15	6, 13, 14	3bitr4r 159	1 |- (E!xph <-> E.x{x | ph} = {x})

Syntax hints:   <-> wb 127  A.wal 672  E.wex 678   = weq 797  E!weu 1007  {cab 1090   = wceq 1091   e. wcel 1092  {csn 1808

This theorem is referenced by:  euuni 1954  reucl 1957  mapsn 3269

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-sn 1811

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