Permutation of Indices of Summation

Theorem

 * $\displaystyle \sum_{\Phi \left({i}\right)} a_i = \sum_{\Phi \left({\pi \left({j}\right)}\right)} a_{\pi \left({j}\right)}$

where:
 * $\displaystyle \sum_{\Phi \left({i}\right)} a_i$ denotes the summation over $a_i$ for all $i$ that satisfy the propositional function $\Phi \left({i}\right)$
 * $\pi$ is a permutation on the fiber of truth of $\Phi$
 * the fiber of truth of $\Phi$ is a finite set.

Also known as
The operation of permutation of indices of a summation can be seen referred to as a permutation of the range.

However, as the term range is ambiguous in the literature, and as its use here is not strictly accurate (it is the fiber of truth of $\Phi$, not its range, which is being permuted, its use on is discouraged.