Prove that double negation elimination is a derivated rule

Question

If $B=\{\wedge i, \wedge e_1, \wedge e_2, \vee i_1, \vee i_2, \vee e, \to i, \to e, \neg i, \neg e\}$, how can I prove that $\neg\neg e$ is a derivated rule from $B$ and proof-by-contradiction?

Answer 1

Assume $\neg\neg\psi$. You want to prove $\psi$.

Hint: $\psi$ will be the conclusion of the proof-by-contradiction rule.

The rest of your proof really ought to write itself now.