Use negation introduction on the unwanted existential quantification, and existential quantifier elimination rule with a fresh witness :
1  ∀x. ¬P(x)    premise
2  | ∃x. P(x)   assumption
3  | | P(a)     assumption, a fresh
4  | | ¬P(a)    ∀-elim 1
5  | | ⊥        ¬-elim 3,4
6  | ⊥          ∃-elim 2,3–5
7  ¬(∃x. P(x))  ¬-intro 2–6
The witness name occurs neither in the surrounding assumptions nor in the conclusion of its subproof. This satisfies the eigenvariable condition; it is not an arbitrarily chosen fixed element assumed to witness the existential quantification. The result is .

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