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Theorem sneqrg 3782
Description: Closed form of sneqr 3780. (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 3651 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
21eqeq1d 2291 . . 3  |-  ( x  =  A  ->  ( { x }  =  { B }  <->  { A }  =  { B } ) )
3 eqeq1 2289 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
42, 3imbi12d 311 . 2  |-  ( x  =  A  ->  (
( { x }  =  { B }  ->  x  =  B )  <->  ( { A }  =  { B }  ->  A  =  B ) ) )
5 vex 2791 . . 3  |-  x  e. 
_V
65sneqr 3780 . 2  |-  ( { x }  =  { B }  ->  x  =  B )
74, 6vtoclg 2843 1  |-  ( A  e.  V  ->  ( { A }  =  { B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {csn 3640
This theorem is referenced by:  sneqbg  3783  altopth1  24499  altopth2  24500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sn 3646
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