$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Question

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let $P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$. By Galois correspondence, let $M' \subset L$ be the subfield corresponding to the subgroup $\operatorname {ker} \Phi_{\mathfrak m,M|K}$.

Now, how do I show that $\operatorname {ker} \Phi_{\mathfrak m,M|K}= \operatorname {ker} \Phi_{\mathfrak m,M'|K}$?