A derivation of the law of excluded middle using only the supplied primitive rules is
1  | ¬(A ∨ ¬A)    assumption
2  | | A          assumption
3  | | A ∨ ¬A     ∨-intro-L 2
4  | | ⊥          ¬-elim 3,1
5  | ¬A           ¬-intro 2–4
6  | A ∨ ¬A       ∨-intro-R 5
7  | ⊥            ¬-elim 6,1
8  ¬¬(A ∨ ¬A)     ¬-intro 1–7
9  A ∨ ¬A         ¬¬-elim 8
Both boxes have been discharged, so the final formula has no undischarged assumptions.

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