quantum mechanics - Fetter & Walecka's derivation of second quantised canonical Schrodinger equation for fermions

On page 18, before the occupation number variables for states i and j are changed $n_i \rightarrow n'_i = n_i - 1$ and $n_j \rightarrow n'_j = n_j + 1$ respectively, could we not have rewritten eq. 1.58 as $$i\hbar \frac{\partial}{\partial t} |\Psi(t)\rangle \\= \dots + \sum_{n_1 n_2 \dots n_\infty} \sum_{i