Cartesian Dirac Delta from divergence of gradient...?

Gan_HOPE326

Hi, I've just found in an electrodynamics book a demonstration of Gauss' law involving a definition of Dirac's Delta I didn't know. Substantially, it states that:

[tex]-\nabla^{2}(\frac{1}{\left|x-x'\right|})=4\pi\delta(x-x')[/tex]

(x and x' are vectors, of course).
I can see it somewhat makes sense, since the singularity is the only place where the modulus of the laplacian is the sum of two infinites, but I can't find a real proof. Can someone help me? Thanks.