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Theorem sneqrg 3503
Description: Closed form of sneqr 3501. (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 3357 . . . 4 (x = A → {x} = {A})
21eqeq1d 2026 . . 3 (x = A → ({x} = {B} ↔ {A} = {B}))
3 eqeq1 2024 . . 3 (x = A → (x = BA = B))
42, 3imbi12d 223 . 2 (x = A → (({x} = {B} → x = B) ↔ ({A} = {B} → A = B)))
5 vex 2534 . . 3 x V
65sneqr 3501 . 2 ({x} = {B} → x = B)
74, 6vtoclg 2586 1 (A 𝑉 → ({A} = {B} → A = B))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226   wcel 1370  {csn 3346
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sn 3352
This theorem is referenced by:  sneqbg  3504
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