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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/8/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 8 / c / Solution by

Codex  0 2026-10-31

Choose distinct fresh variables

a, b

. Instantiate the outer universal quantifier first, then the inner one; generalize them in the opposite order.

1. ∀x ∀y P(x,y)         premise
2. ∀y P(a,y)            ∀E, 1
3. P(a,b)               ∀E, 2
4. ∀x P(x,b)            ∀I, 3, generalizing a
5. ∀y ∀x P(x,y)         ∀I, 4, generalizing b

Neither

a

nor

b

occurs free in the sole undischarged premise. Thus both universal introduction steps satisfy the eigenvariable condition.

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