Theorem sneqrg 3533

Description: Closed form of sneqr 3531. (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 3386	. . . 4 |- ( x = A -> { x } = { A } )
2	1	eqeq1d 2048	. . 3 |- ( x = A -> ( { x } = { B } <-> { A } = { B } ) )
3		eqeq1 2046	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
4	2, 3	imbi12d 223	. 2 |- ( x = A -> ( ( { x } = { B } -> x = B ) <-> ( { A } = { B } -> A = B ) ) )
5		vex 2560	. . 3 |- x e. _V
6	5	sneqr 3531	. 2 |- ( { x } = { B } -> x = B )
7	4, 6	vtoclg 2613	1 |- ( A e. V -> ( { A } = { B } -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sn 3381

This theorem is referenced by:  sneqbg  3534