Let $A$, $B$, $C$ be subsets of a finite set $X$. Let $A \Delta B$ denote the symmetric difference of $A$ and $B$, i.e. the set of elements of $X$ lying in exactly one of $A$ and $B$. Then $A \Delta B$ = $C \Delta D$ if and only if $A \Delta C = B \Delta D$.
This has a one-line proof: take the symmetric difference of both sides of either equation with $B \Delta C$. Underlying this is the fact that the set of all subsets of $X$ is an elementary abelian group of order $2^{|X|}$ under symmetric difference, with the empty set as the identity.