linguistic:paraphrasing, linguistic:translation
Hilbert-style calculus

Intuitionistic logic can be defined using the following Hilbert-style calculus. This is similar to a way of axiomatizing classical propositional logic.

In propositional logic, the inference rule is modus ponens

and the axioms are

To make this a system of first-order predicate logic, the generalization rules

are added, along with the axioms