  Using functional methods and the exact renormalization group we derive Ward
identities for the Anderson impurity model. In particular, we present a
non-perturbative proof of the Yamada-Yosida identities relating certain
coefficients in the low-energy expansion of the self-energy to thermodynamic
particle number and spin susceptibilities of the impurity. Our proof underlines
the relation of the Yamada-Yosida identities to the U(1) x U(1) symmetry
associated with particle number and spin conservation in a magnetic field.
