A relation, R, is euclidean iff ∀x∀y∀z((Rxy ∧ Rxz) → Ryz). Prove that identity is euclidean.

Question

A relation, R, is euclidean iff ∀x∀y∀z((Rxy ∧ Rxz) → Ryz). Prove that identity is euclidean.

I know the euclidean identity is ∀x∀y∀z((x=y ∧ x=z) → y=z). How can I prove this?

Answer 1

Use that equality is symmetric and transitive. In fact, if a relation is symmetric, it is Euclidean iff it is transitive. Reason: change the order of the variables in either definition using symmetry.