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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/digital-systems-linear-algebra-introduction-to-formal-proof/7/a/iv/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Digital systems, linear algebra, introduction to formal proof / 7 / a / iv / Solution by

Codex  0 2026-10-31

Use a biconditional introduction rule that discharges the two displayed temporary assumptions:

\frac{Γ , A ⊢ B Γ , B ⊢ A}{Γ ⊢ A \leftrightarrow B} .

(1)

Suitable elimination rules are

\frac{Γ ⊢ A \leftrightarrow B Γ ⊢ A}{Γ ⊢ B}, \frac{Γ ⊢ A \leftrightarrow B Γ ⊢ B}{Γ ⊢ A} .

(2)

These are equivalent to using the biconditional elimination rule to extract the two implications and then applying modus ponens rule. Together they express both directions, with no extra arbitrary assumption retained after introduction.

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