You can find a proof in Chapter VIII of Cassels-Fröhlich (eds.): Algebraic number theory (Academic Press, 1967). The chapter was written by Heilbronn, and his proof uses the Wiener-Ikehara Tauberiantauberian theorem. See Theorem 4 there.
See also Lagarias-Odlyzko: Effective versions of the Chebotarev density theorem, In: Algebraic number fields: L-functions and Galois properties, pp. 409–464, Academic Press, London, 1977.