We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.
"Intuitionistic Completeness and Classical Logic." Notre Dame J. Formal Logic 43 (4) 243 - 248, 2002. https://doi.org/10.1305/ndjfl/1074396309