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Theorem sneqrg 4042
Description: Closed form of sneqr 4040. (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 3887 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21eqeq1d 2451 . . 3  |-  ( x  =  A  ->  ( { x }  =  { B }  <->  { A }  =  { B } ) )
3 eqeq1 2449 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
42, 3imbi12d 320 . 2  |-  ( x  =  A  ->  (
( { x }  =  { B }  ->  x  =  B )  <->  ( { A }  =  { B }  ->  A  =  B ) ) )
5 vex 2975 . . 3  |-  x  e. 
_V
65sneqr 4040 . 2  |-  ( { x }  =  { B }  ->  x  =  B )
74, 6vtoclg 3030 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {csn 3877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2974  df-sn 3878
This theorem is referenced by:  sneqbg  4043  altopth1  27996  altopth2  27997
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