Theorem reusn 2892

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 1651	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		eusn 2446	. 2 |- (E!x(x e. A /\ ph) <-> E.x{x | (x e. A /\ ph)} = {x})
3		df-rab 1652	. . . . 5 |- {x e. A | ph} = {x | (x e. A /\ ph)}
4	3	eqeq1i 1482	. . . 4 |- ({x e. A | ph} = {x} <-> {x | (x e. A /\ ph)} = {x})
5	4	exbii 1051	. . 3 |- (E.x{x e. A | ph} = {x} <-> E.x{x | (x e. A /\ ph)} = {x})
6		ax-17 971	. . . 4 |- ({x e. A | ph} = {x} -> A.y{x e. A | ph} = {x})
7		hbrab1 1772	. . . . 5 |- (z e. {x e. A | ph} -> A.x z e. {x e. A | ph})
8		ax-17 971	. . . . 5 |- (z e. {y} -> A.x z e. {y})
9	7, 8	hbeq 1565	. . . 4 |- ({x e. A | ph} = {y} -> A.x{x e. A | ph} = {y})
10		sneq 2417	. . . . 5 |- (x = y -> {x} = {y})
11	10	eqeq2d 1486	. . . 4 |- (x = y -> ({x e. A | ph} = {x} <-> {x e. A | ph} = {y}))
12	6, 9, 11	cbvex 1166	. . 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 956   e. wcel 958  E.wex 980  E!weu 1380  {cab 1463  E!wreu 1647  {crab 1648  {csn 2409

This theorem is referenced by:  reusni 2893  reuunisn 2895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-clab 1464  df-cleq 1469  df-clel 1472  df-reu 1651  df-rab 1652  df-sn 2412

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