Kleene recursion theorem [interactive version]:

("Y)($r) Y(r,x) = fr(x)

("z)("k)($r) Yz(k,x) = fr(x)
("z)("k)Yz(k,x) = fs(z,k)(x)
Y(s(k,k),x) = Yd(k,x)
("k)Yd(k,x) = fs(d,k)(x)
Y(s(d,d),x) = Yd(d,x)
Yd(d,x) = fs(d,d)(x)
r = s(d,d)
d = d(z)

Adapted from

Mechanism, Mentalism and Metamathematics - An Essay on Finitism,
Judson C. Webb, D. Reidel Publ. Co. 1980, p. 214.