Theorem sneqrg 3850

Description: Closed form of sneqr 3848. (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 3684	. . . 4 |- ( x = A -> { x } = { A } )
2	1	eqeq1d 2240	. . 3 |- ( x = A -> ( { x } = { B } <-> { A } = { B } ) )
3		eqeq1 2238	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
4	2, 3	imbi12d 234	. 2 |- ( x = A -> ( ( { x } = { B } -> x = B ) <-> ( { A } = { B } -> A = B ) ) )
5		vex 2806	. . 3 |- x e. _V
6	5	sneqr 3848	. 2 |- ( { x } = { B } -> x = B )
7	4, 6	vtoclg 2865	1 |- ( A e. V -> ( { A } = { B } -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   {csn 3673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sn 3679

This theorem is referenced by:  sneqbg  3851