Proving a PDA with CFG Transitions recognizes Context Free Languages I'm working on a proof that deals with a modified PDA, which is identitical to a PDA, but with transitions: \begin{equation*}
a,b \to c
\end{equation*} where $a$ is a context free grammar, instead of just a symbol. The modified PDA then reads a prefix $w'$ of $w$, the suffix of the input stream, where $w' \in L(a)$.

I'm try to prove that the language recognized by this modified PDA is context free, but I'm really stuck. I have no idea how to even get started. If someone could point me in the right direction, I would really appreciate it.
Hint: Try replacing each such transition by an appropriate PDA.