My answer is similar to the one of Dave Tweed, meaning that I put it on a more formal level. I obviously answered later, but I decided to nevertheless post it since someone may find this approach interesting.
The relation you are trying to prove is independent from the structure of the function since it is, as a matter of fact, a tautology. To explain what I mean, I propose a demonstration for a general, correctly formed, Boolean expression in an arbitrary number of Boolean variables, say , , where for all .
We have that and consider the following two sets of Boolean values for the -dimensional Boolean vector
These set are a partition of the full set of values the input Boolean vector can assume, i.e. and (the empty set), thus
therefore we always have