I am having some difficulties proving that if $G/\gamma_2(G)$ is finitely generated, then $\gamma_i(G)/\gamma_{i+1}(G)$ is finitely generated. This is a well-known fact.
Assume that $G/\gamma_2(G)=\langle\overline{x_1},...,\overline{x_n}\rangle$. Further assume that $G=\langle Y\rangle$.
I know that $$\gamma_i(G)/\gamma_{i+1}(G)=\langle[\overline{y_1},...,\overline{y_i}]: y_j\in Y \rangle.$$
I belive it is also true that $$\gamma_i(G)/\gamma_{i+1}(G)=\langle[\overline{y_1},...,\overline{y_i}]: y_j\in \{x_1,...,x_n\} \rangle,$$ but I don't know how to show this. Could someone give me some clue on how to startshowing this equality?
Thank you!