Theorem sneqrg 3875

Description: Closed form of sneqr 3873. (Contributed by Scott Fenton, 1-Apr-2011.)

Assertion

Ref	Expression
sneqrg	|- (A e. V -> ({A} = {B} -> A = B))

Proof of Theorem sneqrg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3745	. . . 4 |- (x = A -> {x} = {A})
2	1	eqeq1d 2361	. . 3 |- (x = A -> ({x} = {B} <-> {A} = {B}))
3		eqeq1 2359	. . 3 |- (x = A -> (x = B <-> A = B))
4	2, 3	imbi12d 311	. 2 |- (x = A -> (({x} = {B} -> x = B) <-> ({A} = {B} -> A = B)))
5		vex 2863	. . 3 |- x e. _V
6	5	sneqr 3873	. 2 |- ({x} = {B} -> x = B)
7	4, 6	vtoclg 2915	1 |- (A e. V -> ({A} = {B} -> A = B))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sn 3742

This theorem is referenced by:  sneqbg  3876