Theorem eusn 2444

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 1568	. . . 4 |- ({x | ph} = {y} <-> A.x(ph <-> x e. {y}))
2		elsn 2419	. . . . . 6 |- (x e. {y} <-> x = y)
3	2	bibi2i 607	. . . . 5 |- ((ph <-> x e. {y}) <-> (ph <-> x = y))
4	3	albii 998	. . . 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 1050	. 2 |- (E.y{x | ph} = {y} <-> E.yA.x(ph <-> x = y))
7		ax-17 970	. . 3 |- ({x | ph} = {x} -> A.y{x | ph} = {x})
8		hbab1 1466	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
9		ax-17 970	. . . 4 |- (z e. {y} -> A.x z e. {y})
10	8, 9	hbeq 1564	. . 3 |- ({x | ph} = {y} -> A.x{x | ph} = {y})
11		sneq 2415	. . . 4 |- (x = y -> {x} = {y})
12	11	eqeq2d 1485	. . 3 |- (x = y -> ({x | ph} = {x} <-> {x | ph} = {y}))
13	7, 10, 12	cbvex 1165	. 2 |- (E.x{x | ph} = {x} <-> E.y{x | ph} = {y})
14		df-eu 1382	. 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 953   = wceq 955   e. wcel 957  E.wex 979  E!weu 1380  {cab 1463  {csn 2407

This theorem is referenced by:  euuni 2878  reucl 2882  reusn 2889  args 3425  mapsn 4342

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-clab 1464  df-cleq 1469  df-clel 1472  df-sn 2410

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