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Theorem sneqrg 3819
Description: Closed form of sneqr 3817. (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 3685 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21eqeq1d 2324 . . 3  |-  ( x  =  A  ->  ( { x }  =  { B }  <->  { A }  =  { B } ) )
3 eqeq1 2322 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
42, 3imbi12d 311 . 2  |-  ( x  =  A  ->  (
( { x }  =  { B }  ->  x  =  B )  <->  ( { A }  =  { B }  ->  A  =  B ) ) )
5 vex 2825 . . 3  |-  x  e. 
_V
65sneqr 3817 . 2  |-  ( { x }  =  { B }  ->  x  =  B )
74, 6vtoclg 2877 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   {csn 3674
This theorem is referenced by:  sneqbg  3820  altopth1  24885  altopth2  24886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sn 3680
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