Theorem reusni 2893

Description: Restricted existential uniqueness determined by a singleton.

Hypothesis

Ref	Expression
reusni.1	|- B e. V

Assertion

Ref	Expression
reusni	|- ({x e. A | ph} = {B} -> E!x e. A ph)

Proof of Theorem reusni

Step	Hyp	Ref	Expression
1		reusni.1	. . 3 |- B e. V
2		sneq 2417	. . . 4 |- (y = B -> {y} = {B})
3	2	eqeq2d 1486	. . 3 |- (y = B -> ({x e. A | ph} = {y} <-> {x e. A | ph} = {B}))
4	1, 3	cla4ev 1869	. 2 |- ({x e. A | ph} = {B} -> E.y{x e. A | ph} = {y})
5		reusn 2892	. 2 |- (E!x e. A ph <-> E.y{x e. A | ph} = {y})
6	4, 5	sylibr 200	1 |- ({x e. A | ph} = {B} -> E!x e. A ph)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  E.wex 980  E!wreu 1647  {crab 1648  Vcvv 1811  {csn 2409

This theorem is referenced by:  rabsnt 2894

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-v 1812  df-sn 2412

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