Littlewood proving that $Li(x) - \pi(x)$ changes sign infinitely often by assuming the Riemann Hypothesis. But later he was able to prove it assuming the Riemann Hypothesis is false.
I find it interesting that you could make opposite assumptions and still arrive at the same result.
What are the examples of other results which have been proved assuming a conjecture to be true and also assuming the same conjecture to be false.