Doubt in Shor-Preskill's proof of security proof of BB84

Question

I was going through the proof of security of the modified Lo-Chau protocol, which eventually leads to the proof of security of BB84. In the error correction step, Nielson-Chuang mention that any bipartite state $\rho^{AB}$, with the promise that there are at most $t$ Pauli errors on the system $B$, can be corrected by measuring $g_i\otimes g_i$, where $g_i$ ranges over the set of generators $\langle g_1, g_2, \ldots, g_{n-m}\rangle$. When I say corrected I mean that they can distill the $m$ EPR states $|\Phi\rangle^{\otimes m}$. Of course Alice and Bob have to do this by measuring $g_i$ locally and then using classical communication. My question is, is there a proof of this claim? Nielsen-Chaung provide an exercise, Exercise 12.34, but I am having a hard time with this. Any help is appreciated.