Negative translation of a classical proof ($\Gamma {⊢}_{\mathrm{CL}}A\Rightarrow {\Gamma }^N{⊢}_{\mathrm{IL}}{A}^N$)

The Gödel-Gentzen negative translation transforms $\Gamma {⊢}_{\mathrm{CL}}A$ into ${\Gamma }^N{⊢}_{\mathrm{IL}}{A}^N$. Induct on a natural deduction derivation. The translated logical conjunction, logical implication and universal rules are ordinary intuitionistic rules. Logical disjunction and existential introduction add double logical negation; their elimination rules first derive the double logical negation of the stable translated conclusion, then use its stability. Classical double-negation elimination translates precisely to stability. If logical equality atoms are double-negated, transport a stable translated conclusion through a logical equality by contradiction and then remove the resulting double logical negation. Original nonlogical axioms must also be translated; the result does not say arbitrary untranslated classical first-order theories are intuitionistically valid.