Frattini subgroup is finitely generated implies subset is generating set iff image in Frattini quotient is

Statement with symbols
Suppose $$G$$ is a group, and $$\Phi(G)$$ (the Frattini subgroup of $$G$$) is a finitely generated group. Then, a subset $$S$$ of $$G$$ is a generating set for $$G$$ if and only if the image of $$S$$ in the Frattini quotient $$G/\Phi(G)$$ under the quotient map, generates $$G/\Phi(G)$$.