PHP_n is a tautology for each n and hence provable in any complete proof system for propositional logic

as the set of formulas derivable from some set of axioms of \(\Gamma_{\mathcal{L}}\) rather than the class of formulas true in all structures – the validity problem is understood to coincide with the problem of deciding whether \(\phi\) is derivable from \(\Gamma_{\mathcal{L}}\). In such cases, the satisfiability and model checking problems are generally not considered. The problems \(\sc{SATISFIABILITY}_{\mathcal{L}}\), \(\sc{VALIDITY}_{\mathcal{L}}\), and \(\sc{MODEL}\ \sc{CHECKING}_{\mathcal{