I have given the answer below with each step explained, but here's an English-language explanation.
The rule of modus tollens allows us to conclude right away that two things are false:
First, it is not the case that not E.
Second, it is not the case that (L if and only if N)
That second inference, by the rule of disjunctive syllogism, implies that the other disjunct on line 2 must be true: if P then not E.
However, we already figured out that "not E" is a false statement. Therefore, again by modus tollens, P cannot be the case.
1. (L ↔️ N) > C
2. (L ↔️ N) v (P > ~E)
3. ~E > C
4. ~C / ~P
5 ~~E MT 3, 4
6 ~(L ↔️ N) MT 1, 4
7 P > ~E DS 2, 6
8 ~P MT 5, 7 QED