Although not exactly what you're after, the question reminds me of Reynolds' parametricity theorem, or as Philip Wadler puts it: Theorems for Free!
The basic idea is that a polymorphic construction (in a polymorphic lambda calculus) must behave uniformly, and so must preserve relations. For example, any term of type $\Pi X. X\to X$ must be the identity function, and every term of type $\Pi X Y. X\times Y\to X$ must be the first projection.
