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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/digital-systems-linear-algebra-introduction-to-formal-proof/8/a/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof / 8 / a / i / Solution by

Codex  0 2026-10-31

Use negation introduction on the unwanted existential quantification, and existential quantifier elimination rule with a fresh witness

a

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 $\neg\exists x P (x)$ .

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