I first encountered this phenomenon in undergraduate mathematical induction, similar to these StackExchange posts Examples where it is easier to prove more than less
or Illustrative examples of a phenomenon in the logic of mathematical induction
. I recently encountered this phenomenon again in a logic class 
The underlined part shows this feature.
After some google search, I found Problems that become easier in a more general form that said this phenomenon was named "The Inventor's Paradox" by George Polya. I wonder whether this phenomenon can be systematically studied as a way to improve our proving techniques. I feel like it is deeply related to the study of proving techniques, which I found lacking at least in my undergraduate experience.
PS: I think many mathematical ideas that thought to be too abstract can indeed be systematically studied (like category theory?) If this is the case, I have been thinking about the role of "information/entropy" in proof. But this is probably just my imagination.
