"Frege's" proof:

(1) | ~Ax~AyRxy A 
(2) | | Ax~Rxb A 
(3) | | ~Rab 2 AE 
(4) | | | AyRay A 
(5) | | | Rab 4 AE 
(6) | | ~AyRay 4,3,5 RAA 
(7) | | Ax~AyRxy 6 AI 
(8) | ~Ax~Rxb 2,1,7 RAA 
(9) | Ay~Ax~Rxy 8 AI 
(10) ~Ax~AyRxy -> Ay~Ax~Rxy 1,9 ->I 

Dan's version:

	1	~ALL(x):~ALL(y):R(x,y)
		Premise

		2	ALL(x):~R(x,b)
			Premise

		3	~R(a,b)
			U Spec, 2

			4	ALL(y):R(a,y)
				Premise

			5	R(a,b)
				U Spec, 4

			6	R(a,b) & ~R(a,b)
				Join, 5, 3

		7	~ALL(y):R(a,y)
			Conclusion, 4

		8	ALL(x):~ALL(y):R(x,y)
			U Gen, 7

		9	ALL(x):~ALL(y):R(x,y) & ~ALL(x):~ALL(y):R(x,y)
			Join, 8, 1

	10	~ALL(x):~R(x,b)
		Conclusion, 2

	11	ALL(y):~ALL(x):~R(x,y)
		U Gen, 10

12	~ALL(x):~ALL(y):R(x,y) => ALL(y):~ALL(x):~R(x,y)
	Conclusion, 1