Theorem sneqrg 3554

Description: Closed form of sneqr 3552. (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.

Step	Hyp	Ref	Expression
1		sneq 3409	. . . 4 |- ( x = A -> { x } = { A } )
2	1	eqeq1d 2089	. . 3 |- ( x = A -> ( { x } = { B } <-> { A } = { B } ) )
3		eqeq1 2087	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
4	2, 3	imbi12d 232	. 2 |- ( x = A -> ( ( { x } = { B } -> x = B ) <-> ( { A } = { B } -> A = B ) ) )
5		vex 2604	. . 3 |- x e. _V
6	5	sneqr 3552	. 2 |- ( { x } = { B } -> x = B )
7	4, 6	vtoclg 2658	1 |- ( A e. V -> ( { A } = { B } -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {csn 3398

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by:  sneqbg  3555