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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.
StepHypRef Expression
1 sneq 3744 . . . 4 (x = A → {x} = {A})
21eqeq1d 2361 . . 3 (x = A → ({x} = {B} ↔ {A} = {B}))
3 eqeq1 2359 . . 3 (x = A → (x = BA = B))
42, 3imbi12d 311 . 2 (x = A → (({x} = {B} → x = B) ↔ ({A} = {B} → A = B)))
5 vex 2862 . . 3 x V
65sneqr 3872 . 2 ({x} = {B} → x = B)
74, 6vtoclg 2914 1 (A V → ({A} = {B} → A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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