Theorem sneqrgOLD 4373

Description: Obsolete proof of sneqrg 4370 as of 23-Jul-2021. (Contributed by Scott Fenton, 1-Apr-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sneqrgOLD	|- ( A e. V -> ( { A } = { B } -> A = B ) )

Proof of Theorem sneqrgOLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4187	. . . 4 |- ( x = A -> { x } = { A } )
2	1	eqeq1d 2624	. . 3 |- ( x = A -> ( { x } = { B } <-> { A } = { B } ) )
3		eqeq1 2626	. . 3 |- ( x = A -> ( x = B <-> A = B ) )
4	2, 3	imbi12d 334	. 2 |- ( x = A -> ( ( { x } = { B } -> x = B ) <-> ( { A } = { B } -> A = B ) ) )
5		vex 3203	. . 3 |- x e. _V
6	5	sneqr 4371	. 2 |- ( { x } = { B } -> x = B )
7	4, 6	vtoclg 3266	1 |- ( A e. V -> ( { A } = { B } -> A = B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by: (None)