Rules for Manipulating Summations

Theorem
Let $R: \Z \to \set {\T, \F}$ and $S: \Z \to \set {\T, \F}$ and so on be propositional functions on the set of integers $\Z$.

Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.

Let $\pi$ be a permutation on the fiber of truth of $R$.