Theorem eusn 2436

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		abeq1 1561	. . . 4 |- ({x | ph} = {y} <-> A.x(ph <-> x e. {y}))
2		elsn 2411	. . . . . 6 |- (x e. {y} <-> x = y)
3	2	bibi2i 606	. . . . 5 |- ((ph <-> x e. {y}) <-> (ph <-> x = y))
4	3	albii 996	. . . 4 |- (A.x(ph <-> x e. {y}) <-> A.x(ph <-> x = y))
5	1, 4	bitr 173	. . 3 |- ({x | ph} = {y} <-> A.x(ph <-> x = y))
6	5	exbii 1047	. 2 |- (E.y{x | ph} = {y} <-> E.yA.x(ph <-> x = y))
7		ax-17 968	. . 3 |- ({x | ph} = {x} -> A.y{x | ph} = {x})
8		hbab1 1459	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
9		ax-17 968	. . . 4 |- (z e. {y} -> A.x z e. {y})
10	8, 9	hbeq 1557	. . 3 |- ({x | ph} = {y} -> A.x{x | ph} = {y})
11		sneq 2407	. . . 4 |- (x = y -> {x} = {y})
12	11	eqeq2d 1478	. . 3 |- (x = y -> ({x | ph} = {x} <-> {x | ph} = {y}))
13	7, 10, 12	cbvex 1162	. 2 |- (E.x{x | ph} = {x} <-> E.y{x | ph} = {y})
14		df-eu 1375	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
15	6, 13, 14	3bitr4r 184	1 |- (E!xph <-> E.x{x | ph} = {x})

Syntax hints:   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  {cab 1456  {csn 2399

This theorem is referenced by:  euuni 2871  reucl 2875  reusn 2882  args 3412  mapsn 4329

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-clab 1457  df-cleq 1462  df-clel 1465  df-sn 2402

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