I use the Fitch notation for the natural deduction system. More information on https://en.wikipedia.org/wiki/Fitch_notation.
In attempting to derive "$P \iff ¬P$" without any previous assumptions, I can't seem to get anywhere.
My 1st approach is to indirectly prove this (IP), so the proof would begin as a sub-proof $¬(P \iff ¬P)$. but I can't seem to arrive at any sort of useful contradiction.
My second approach is directly proving that $P \Rightarrow ¬P$ and $¬P \Rightarrow P$, but getting the negation of either seems like a dead-end.