The \(\varepsilon \)-elimination method of Hilbert’s \(\varepsilon \)-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s \(\varepsilon \)-calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses of the extended first \(\varepsilon \)-theorem, even if the formalisation incorporates so-called \(\varepsilon \)-equality axioms.

希尔伯特\(\varepsilon \)微积分的\(\varepsilon \)消去法产生了用于计算外延公式的Herbrand 析取的最新最直接算法。一个核心优点是获得的 Herbrand 复杂度的上限与证明的命题结构无关。希尔伯特微积分的先前（现代）工作主要集中于纯微积分，没有等式。我们澄清这种独立性也适用于具有相等性的一阶逻辑。此外，我们提供了扩展的第一\(\varepsilon \)定理的上限分析，即使形式化包含所谓的\(\varepsilon \)等式公理。