Theorem sneqrg 3874

Description: Closed form of sneqr 3872. (Contributed by Scott Fenton, 1-Apr-2011.)

Assertion

Ref	Expression
sneqrg	⊢ (A ∈ V → ({A} = {B} → A = B))

Proof of Theorem sneqrg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3744	. . . 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 2862	. . 3 ⊢ x ∈ V
6	5	sneqr 3872	. 2 ⊢ ({x} = {B} → x = B)
7	4, 6	vtoclg 2914	1 ⊢ (A ∈ V → ({A} = {B} → A = B))

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {csn 3737

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 2478  df-v 2861  df-sn 3741

This theorem is referenced by:  sneqbg  3875