Theorem reusn 3818

Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)

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 2111	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		euabsn 3095	. 2 |- (E!x(x e. A /\ ph) <-> E.x{x | (x e. A /\ ph)} = {x})
3		ax-17 1317	. . 3 |- ({x | (x e. A /\ ph)} = {x} -> A.y{x | (x e. A /\ ph)} = {x})
4		hbrab1 2257	. . . 4 |- (z e. {x e. A | ph} -> A.x z e. {x e. A | ph})
5		ax-17 1317	. . . 4 |- (z e. {y} -> A.x z e. {y})
6	4, 5	hbeq 1995	. . 3 |- ({x e. A | ph} = {y} -> A.x{x e. A | ph} = {y})
7		df-rab 2112	. . . . . 6 |- {x e. A | ph} = {x | (x e. A /\ ph)}
8	7	eqcomi 1888	. . . . 5 |- {x | (x e. A /\ ph)} = {x e. A | ph}
9	8	a1i 8	. . . 4 |- (x = y -> {x | (x e. A /\ ph)} = {x e. A | ph})
10		sneq 3054	. . . 4 |- (x = y -> {x} = {y})
11	9, 10	eqeq12d 1899	. . 3 |- (x = y -> ({x | (x e. A /\ ph)} = {x} <-> {x e. A | ph} = {y}))
12	3, 6, 11	cbvex 1529	. 2 |- (E.x{x | (x e. A /\ ph)} = {x} <-> E.y{x e. A | ph} = {y})
13	1, 2, 12	3bitri 194	1 |- (E!x e. A ph <-> E.y{x e. A | ph} = {y})

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  E!wreu 2107  {crab 2108  {csn 3044

This theorem is referenced by:  reusni 3820  reuunisn 3822

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-reu 2111  df-rab 2112  df-sn 3049

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