Equivalence of definitions of verbal subgroup

Statement
The following are equivalent for a subgroup $$H$$ of a group $$G$$:


 * 1) There is a collection $$C_1$$ of words such that $$H$$ is precisely the span of $$C_1$$, i.e., $$H$$ is the union, over all words in $$C_1$$, of the images of the corresponding word maps.
 * 2) There is a collection $$C_2$$ of words such that $$H$$ is precisely the subgroup generated by the span of $$C_2$$, i.e., $$H$$ is the subgroup generated by the union, over all words in $$C_2$$, of the images of the corresponding word maps.
 * 3) There is a subvariety $$\mathcal{V}$$ of the variety of groups such that $$H$$ is the unique smallest normal subgroup of $$G$$ for which $$G/H \in \mathcal{V}$$.

A subgroup $$H$$ satisfying these equivalent conditions is termed a verbal subgroup.

(1) implies (2)
We could take $$C_2 = C_1$$.

(2) implies (1)
We take $$C_1$$ as the collection of all words that can be written as words in terms of the words of $$C_2$$. This is a form of composition of words.

(2) implies (3)
We take $$\mathcal{V}$$ as the variety defined by equations saying that each word of $$C_2$$ equals the identity element.

(3) implies (2)
We need to use the Birkhoff-von Neumann theorem which asserts that any variety can be defined equationally. So, we find a bunch of equations describing $$\mathcal{V}$$. Rewrite those equations by bringing everything to one side, and define $$C_2$$ as the set of the words corresponding to these equations.