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Theorem sneqrg 3524
Description: Closed form of sneqr 3522. (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 3378 . . . 4 (x = A → {x} = {A})
21eqeq1d 2045 . . 3 (x = A → ({x} = {B} ↔ {A} = {B}))
3 eqeq1 2043 . . 3 (x = A → (x = BA = B))
42, 3imbi12d 223 . 2 (x = A → (({x} = {B} → x = B) ↔ ({A} = {B} → A = B)))
5 vex 2554 . . 3 x V
65sneqr 3522 . 2 ({x} = {B} → x = B)
74, 6vtoclg 2607 1 (A 𝑉 → ({A} = {B} → A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sn 3373
This theorem is referenced by:  sneqbg  3525
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