Prove that the minimum number of generators of a finite $p$-group $G$ is $\dim_{\mathbb{Z}/p\mathbb{Z}}(G/G^p[G,G])$, the dimension of the vector space $\mathbb{Z}/p\mathbb{Z}$ over $\mathbb{Z}/p\mathbb{Z}$
I know that if $G$ is a finite $p$-group, then $G^p[G,G] = \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$, and that if $V$ is a vector space over $\mathbb{Z}/p\mathbb{Z}$, then $\Phi(V) = \{0\}$, so we would have $$\Phi\left(G/\Phi(G)\right) = \{0\}$$
I also know that the Frattini subgroup of $G$ is the set of non-generators of $G$, i.e., $$d\left(G\right) = d\left(G/\Phi(G)\right)$$ where $d()$ dentoes the minimum number of generators. But I'm not sure how to put this all together to prove what I need to prove. Thanks.