I got a question regarding the decidability of equivalence of two context free grammars:
Construct a Turing machine that decides whether $L(G) = L(H)$, where $G$ and $H$ are two context free grammars.
This question is taken from Sipser's book on theory of computation.
My current idea is given that $G$ and $H$ are CFG's, we know that there exists push down automata that accept the languages described by $G$ and $H$. We can simulate a PDA on a Turing machine and hence, we can convert the problem to the equality of Turing machines, such that $M_1$ and $M_2$ are TMs and $L(M_1) = L(M_2)$ which is known to be undecidable.
My question is, are these steps fine to do (with a bit more formalism)?