Theorem sneqrg 3764

Description: Closed form of sneqr 3762. (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 3605	. . . 4 |- ( x = A -> { x } = { A } )
2	1	eqeq1d 2186	. . 3 |- ( x = A -> ( { x } = { B } <-> { A } = { B } ) )
3		eqeq1 2184	. . 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 2742	. . 3 |- x e. _V
6	5	sneqr 3762	. 2 |- ( { x } = { B } -> x = B )
7	4, 6	vtoclg 2799	1 |- ( A e. V -> ( { A } = { B } -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {csn 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sn 3600

This theorem is referenced by:  sneqbg  3765