I read on the wikipedia article for the Riemann Hypothesis that some theorems have been proved by assuming the hypothesis to be true and then false and proving the certain theorem from both cases. I.e. proving $P\Rightarrow Q$ and $\neg P\Rightarrow Q$ and then infering $Q$.
This method made me curious about another possible method. What if you prove $P \Leftrightarrow Q$ and then $\neg P \Rightarrow Q$, doesn't that prove both $P$ and $Q$? Is that logically sound? If so it would be an interesting proof method because if I proved $P \Leftrightarrow Q$ I would've thought nothing more could be gleaned from the relation of their truth values, but if this is true checking $\neg P\Rightarrow Q$ could prove both theorems.
If this makes any sense are there any examples of any proofs like this?