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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/8/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 8 / a / Solution by

Codex  0 2026-10-25

Let

N

abbreviate the given negated existential statement, and choose a fresh variable

u

that is not free in the undischarged premise. Work from an assumed

P (u)

Q (u)

1  ¬∃x (Q(x) → P(x))                  premise N
2  | P(u)                             assume
3  | | Q(u)                           assume
4  | | P(u)                           available from 2
5  | Q(u) → P(u)                      →-intro, discharge 3–4
6  | ∃x (Q(x) → P(x))                 ∃-intro, witness u in 5
7  | ⊥                                ¬-elim, 6 and 1
8  | Q(u)                             ⊥-elim, 7
9  P(u) → Q(u)                        →-intro, discharge 2–8
10 ∀x (P(x) → Q(x))                   ∀-intro, 9, u fresh

The occurrence of

u

in the temporary assumption at line 2 is harmless because that assumption is discharged before universal quantifier introduction rule. No unproved implication-creation rule or quantifier-negation equivalence is needed.

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