Let's just say that some person discovered that $P = NP$ implies $P \neq NP$ and $P \neq NP$ implies $P = NP$, and we don't know what is causing this contradiction, And this was a valid proof that was accepted. What would the consequences of this be? This would probably be the least likely outcome of the P vs NP question. But one I don't see talked about.
If you have a proof that $(P=NP)\Leftrightarrow (P\not= NP) $ then $P=NP$ is proveable is equivalent to $P\not= NP$ is provable. This would mean that neither $P=NP$ nor $P\not= NP$ is provable.
But if there would be a proof, that $P=NP$ is not provable then it would be proven that one cannot find an efficient algorithm for SAT or any other $NP$-complete problem. This would mean that $P \not= NP$. So, if $P$ versus $NP$ is independent from math then this would not be provable, too.