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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/c/iii/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 / c / iii / Solution by

Codex  0 2026-10-31

Choose distinct fresh parameters

a, b

. Within the assumption

a = b

, equality elimination rule gives the forward implication directly. Derive the reverse equality explicitly before using it for the reverse implication:

1  | a = b                                           assumption
 2  | | P(a)                                         assumption
 3  | | P(b)                                         =-elim 1,2
 4  | P(a) → P(b)                                    →-intro 2–3
 5  | a = a                                          =-intro
 6  | b = a                                          =-elim 1,5; φ(u) := (u = a)
 7  | | P(b)                                         assumption
 8  | | P(a)                                         =-elim 6,7
 9  | P(b) → P(a)                                    →-intro 7–8
10  | (P(a) → P(b)) ∧ (P(b) → P(a))                  ∧-intro 4,9
11  a = b → ((P(a) → P(b)) ∧ (P(b) → P(a)))          →-intro 1–10
12  ∀y. a = y → ((P(a) → P(y)) ∧ (P(y) → P(a)))      ∀-intro 11, b fresh
13  ∀x. ∀y. x = y → ((P(x) → P(y)) ∧ (P(y) → P(x)))  ∀-intro 12, a fresh

The equality assumption is discharged before either universal quantifier introduction rule. There are then no undischarged assumptions, so both eigenvariable condition requirements hold. The final universal theorem expresses that equal arguments are interchangeable in the predicate

P

in both directions.

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