Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃ x : A. F(x) is understood as disjoint union, are the projections, and these do not preserve first-orderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting’s system.
"The completeness of Heyting first-order logic." J. Symbolic Logic 68 (3) 751 - 763, September 2003. https://doi.org/10.2178/jsl/1058448436