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This paper is about the elementary metatheory of equational predicate logic—that is, first‐order logic that uses Leibniz's substitution of ‘equals for equals’ as a primary rule of inference. We present here two different ‘faithful’ formalizations of this logic and prove that both versions are sound and complete. A by‐product of this study is a proof that the ‘full’ Leibniz rule is strictly stronger than the ‘no‐capture’ Leibniz rule. We obtain this result here for a  complete  Logic. We also show that under some reasonable conditions, propositional Leibniz, no‐capture Leibniz, and a full‐capture version are all equivalent, provided that the latter is restricted to act on universally valid premises whenever capture is allowed.

On the Soundness and Completeness of Equational Predicate Logics