Use only the displayed natural deduction rules. Let abbreviate . Indentation below marks temporary assumptions; the closing rule names the assumption discharged.
1  | ¬A                         assume
2  | | φ                        assume
3  | | A                        ∨-intro-L, 2
4  | | ⊥                        ¬-elim, 3 and 1
5  | ¬φ                         ¬-intro, discharge 2–4
6  | A                          ∨-intro-R, 5
7  | ⊥                          ¬-elim, 6 and 1
8  ¬¬A                          ¬-intro, discharge 1–7
9  A                            ¬¬-elim, 8
There are no undischarged assumptions at line 9, so this proves the law of excluded middle. The use of double-negation elimination is explicit; an unproved appeal to the derived RAA rule is unnecessary.

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