A term is called a ternary deductive (TD) term for a variety of algebras if the identity holds in and yields for any and any principal congruence on . A connective is called -distributive if . If is a propositional logic and is a corresponding variety (algebraic semantic) that has a TD term , then any admissible in rule, the premises of which contain only -distributive operations, is derivable, and the substitution is a projective -unifier for any formula containing only -distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic.
"Algebraic Logic Perspective on Prucnal’s Substitution." Notre Dame J. Formal Logic 57 (4) 503 - 521, 2016. https://doi.org/10.1215/00294527-3659423