If the classical predicate logic is formulated in a sequent calculus, say LJP, by adding the so-called Peirce rule to Gentzen's intuitionistic LJ, the commonly known proof form of cut elimination for the system cannot be formalized in primitive recursive arithmetic. The author, first, provides a system called LJP(\(\Delta)\) so that the standard cut elimination arguments for it can be carried out with a slight modification. A primitive recursive cut-elimination procedure for LJP is then obtained by two transformations between them; one from LJP derivations to those of LJP(\(\Delta)\) and the other from cut-free LJP(\(\Delta)\) derivations to those of LJP. The complexity of cut-free derivation thus obtained is also discussed in both the predicate and the propositional case.