Theorem reusn 2887

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

Assertion

Ref	Expression
reusn	|- (E!x e. A ph <-> E.y{x e. A | ph} = {y})

Distinct variable groups:   y,A   ph,y   x,y

Proof of Theorem reusn

Step	Hyp	Ref	Expression
1		df-reu 1648	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		eusn 2442	. 2 |- (E!x(x e. A /\ ph) <-> E.x{x | (x e. A /\ ph)} = {x})
3		df-rab 1649	. . . . 5 |- {x e. A | ph} = {x | (x e. A /\ ph)}
4	3	eqeq1i 1479	. . . 4 |- ({x e. A | ph} = {x} <-> {x | (x e. A /\ ph)} = {x})
5	4	exbii 1049	. . 3 |- (E.x{x e. A | ph} = {x} <-> E.x{x | (x e. A /\ ph)} = {x})
6		ax-17 969	. . . 4 |- ({x e. A | ph} = {x} -> A.y{x e. A | ph} = {x})
7		hbrab1 1769	. . . . 5 |- (z e. {x e. A | ph} -> A.x z e. {x e. A | ph})
8		ax-17 969	. . . . 5 |- (z e. {y} -> A.x z e. {y})
9	7, 8	hbeq 1562	. . . 4 |- ({x e. A | ph} = {y} -> A.x{x e. A | ph} = {y})
10		sneq 2413	. . . . 5 |- (x = y -> {x} = {y})
11	10	eqeq2d 1483	. . . 4 |- (x = y -> ({x e. A | ph} = {x} <-> {x e. A | ph} = {y}))
12	6, 9, 11	cbvex 1164	. . 3 |- (E.x{x e. A | ph} = {x} <-> E.y{x e. A | ph} = {y})
13	5, 12	bitr3 175	. 2 |- (E.x{x | (x e. A /\ ph)} = {x} <-> E.y{x e. A | ph} = {y})
14	1, 2, 13	3bitr 177	1 |- (E!x e. A ph <-> E.y{x e. A | ph} = {y})

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  E!weu 1378  {cab 1461  E!wreu 1644  {crab 1645  {csn 2405

This theorem is referenced by:  reusni 2888  reuunisn 2890

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-clab 1462  df-cleq 1467  df-clel 1470  df-reu 1648  df-rab 1649  df-sn 2408

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