I am amazed by this idea, since to me the fundamental principle of problem solving is to make the problem easier, and i always assumed this meant making it more special.  It is true that proving a theorem is easier by ignoring irrelevant facets, but these are only known after solving the problem.  I find it is more productive in discovering which facets are relevant to do various examples, gradually trying to generalize the argument.  Even Deligne proved the Weil conjectures first for K3 surfaces.