Theorem sneqrg 3792

Description: Closed form of sneqr 3790. (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 3633	. . . 4 |- ( x = A -> { x } = { A } )
2	1	eqeq1d 2205	. . 3 |- ( x = A -> ( { x } = { B } <-> { A } = { B } ) )
3		eqeq1 2203	. . 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 2766	. . 3 |- x e. _V
6	5	sneqr 3790	. 2 |- ( { x } = { B } -> x = B )
7	4, 6	vtoclg 2824	1 |- ( A e. V -> ( { A } = { B } -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3628

This theorem is referenced by:  sneqbg  3793