Among the many mind-blowing things in complexity|theory are the "win-win" theorems. If one can show that "if A doesn't work, well then B must"... we're done.

E.g., "A is efficient if [Hard Conjecture] is true, but B is efficient if it isn't." No need to prove [Hard Conjecture]!

It's even more elegant as the "wins" are the same. Sometimes, you get
- if [X] holds, then [something I want]
- if [X] doesn't, then [something else I want]
but there are many examples where the conclusion is identical in both cases! Just taking a different path to get there...

Incidentally, if you are interested in that sort of arguments in complexity and learning theory, then definitely check out Igor Carboni Oliveira's work ( https://www.dcs.warwick.ac.uk/~igorcarb/ ).