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Theorem sneqrg 3606
Description: Closed form of sneqr 3604. (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.
StepHypRef Expression
1 sneq 3457 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21eqeq1d 2096 . . 3  |-  ( x  =  A  ->  ( { x }  =  { B }  <->  { A }  =  { B } ) )
3 eqeq1 2094 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
42, 3imbi12d 232 . 2  |-  ( x  =  A  ->  (
( { x }  =  { B }  ->  x  =  B )  <->  ( { A }  =  { B }  ->  A  =  B ) ) )
5 vex 2622 . . 3  |-  x  e. 
_V
65sneqr 3604 . 2  |-  ( { x }  =  { B }  ->  x  =  B )
74, 6vtoclg 2679 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sn 3452
This theorem is referenced by:  sneqbg  3607
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