Write for . The following derives Peirce's law directly from the printed primitive rules, including double-negation elimination:
LineFormulaJustification
1Assume for the final implication
2Assume locally
3Assume inside the scope of 2
4Negation elimination rule, 2 and 3
5Logical falsity elimination rule, 4
6Implication introduction rule, discharging 3–5
7Implication elimination rule, 1 and 6
8Negation elimination rule, 2 and 7
9Negation introduction rule, discharging 2–8
10Double-negation elimination, 9
11Implication introduction rule, discharging 1–10
Every assumption is discharged, so the last formula is a theorem. In particular, no unproved appeal to the derived RAA rule is hidden in the argument.

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