Using functional methods and the exact renormalization group we derive Ward identities for the Anderson impurity model. In particular, we present a nonperturbative proof of the Yamada–Yosida identities relating certain coefficients in the low-energy expansion of the self-energy to the thermodynamic particle number and spin susceptibilities of the impurity. Our proof underlines the relation of the Yamada–Yosida identities to the U(1) × U(1) symmetry associated with the particle number and spin conservation in a magnetic field. PACS numbers: 72.15.Qm, 71.27.+q, 71.10.Pm