Suppose $G = \langle x_1, \ldots, x_n \mid p_1, \ldots, p_m \rangle$ is a finitely presented group, and let $\langle A \mid R \rangle$ be another presentation of $G$, with $A$ and $R$ possibly infinite. Do there always exist finite subsets $A' \subset A$, $R' \subset R$ such that $G = \langle A' \mid R' \rangle$?
I feel like the answer should be "yes." Here's my idea: we can write each $x_i$ as a product of finitely many $a \in A$ (and their inverses); denote by $A_i$ this finite set of $a$'s. Then the finite set $A_1 \cup \cdots \cup A_n$ generates $G$.
Similarly, each relator $p_i$ can be derived from a finite set of relators $R_i \subset R$. Here's my problem, though: how do I know that any relator $w$ in the letters $A_1 \cup \cdots \cup A_n$ can be reduced using these $R_i$? Not every such $w$ blocks off into $x_i$-chunks.