Hilbert’s ε-calculus internalizes existential reasoning by means of a choice term εx P (x) satisfying ∃x P (x) → P (εx P (x) ). In general, such a term need not be canonical. This paper isolates the effect of adding uniqueness in a Hilbert-style first-order system with equality and ε. It shows that if P has exactly one witness, then εx P (x) is forced to denote that witness. The paper also gives a corresponding selector-theoretic formulation: for the operator SelP (S): = x ∈ S: P (x) on P (D), the fixed points are classified in general, and under existence and uniqueness the non-empty fixed-point structure collapses to the singleton generated by the canonical witness. These results are presented as an explicit derivation of a structural fact that is often left implicit, together with a machine-checked Coq/Rocq formalization and negative test files showing that the derivation depends on both existence and uniqueness assumptions.
Nived Rajendran (Wed,) studied this question.