Theorem reusni 2899

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 2421	. . . 4 |- (y = B -> {y} = {B})
3	2	eqeq2d 1489	. . 3 |- (y = B -> ({x e. A | ph} = {y} <-> {x e. A | ph} = {B}))
4	1, 3	cla4ev 1872	. 2 |- ({x e. A | ph} = {B} -> E.y{x e. A | ph} = {y})
5		reusn 2898	. 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 958   e. wcel 960  E.wex 982  E!wreu 1650  {crab 1651  Vcvv 1814  {csn 2413

This theorem is referenced by:  rabsnt 2900

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-clab 1467  df-cleq 1472  df-clel 1475  df-reu 1654  df-rab 1655  df-v 1815  df-sn 2416

Copyright terms: Public domain