I came across the following theorem of class field theory in the appendix of the book 'Cyclotomic fields' by Washington. I could not find it in Milne, Cassels Frohlich, Neukirch or Artin Tate.
Theorem 14. (a) If $K/k$ is abelian, then there is a closed subgroup $H$ with $D_k\subseteq H\subseteq C_k$, such that $$C_k/H\simeq\mathrm{Gal}(K/k).$$ The prime $\mathfrak{p}$ is unramified $\Longleftrightarrow k^\times U_\mathfrak{p}/k^\times\subseteq H$.
(b) Given a closed subgroup $H$ with $D_k\subseteq H\subseteq C_k$ (equivalently, $C_k/H$ is totally disconnected), there is a unique abelian extension corresponding to $H$, as in (a).
Where can I find proof and discussion on this theorem?