Commutator of normal subgroups equals normal closure of commutators of generators

Statement
Suppose $$G$$ is a group and $$H,K$$ are normal subgroups of $$G$$. Suppose $$A$$ is a generating set for $$H$$ and $$B$$ is a generating set for $$K$$. Then, the commutator subgroup $$[H,K]$$ is the normal closure of the subgroup:

$$\langle [a,b] \mid a \in A, b \in B \rangle$$

In other words, it is the smallest normal subgroup containing all the commutators between elements of $$A$$ and elements of $$B$$.