Quotient of finite, nilpotent group by Frattini subgroup is isomorphic to product of quotients of Sylow subgroups by their respective Frattini groups.

Question

Let $G$ be a finite nilpotent group.

We know that $G=G_{p_1}\times G_{p_2}\times \cdots \times G_{p_r}$ where $G_{p_i}\in Syl_{p_i}(G)$, $i=1,\dots,r$.

Is the following equation right? And why?

$G/\Phi(G)=G_{p_1}/\Phi(G_{p_1})\times\cdots\times G_{p_r}/\Phi(G_{p_r})$,

where $\Phi(G)$ is the Frattini subgroup of $G$.