From forcing to satisfaction in Kripke models of intuitionistic predicate logic

We answer a natural question concerning the relation between forcing and satisfaction in Kripke models of intuitionistic predicate logic. We define a class of formulas denoted by R∗, with the property that forcing of any R∗-formula in a node of a Kripke model of intuitionistic predicate logic implies its satisfaction in the classical structure attached to that node. We also prove that any formula with this property is an R∗-formula.