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Theorem sneqrg 3689
Description: Closed form of sneqr 3687. (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 3538 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21eqeq1d 2148 . . 3  |-  ( x  =  A  ->  ( { x }  =  { B }  <->  { A }  =  { B } ) )
3 eqeq1 2146 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
42, 3imbi12d 233 . 2  |-  ( x  =  A  ->  (
( { x }  =  { B }  ->  x  =  B )  <->  ( { A }  =  { B }  ->  A  =  B ) ) )
5 vex 2689 . . 3  |-  x  e. 
_V
65sneqr 3687 . 2  |-  ( { x }  =  { B }  ->  x  =  B )
74, 6vtoclg 2746 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   {csn 3527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sn 3533
This theorem is referenced by:  sneqbg  3690
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