A On the Power of Substitution in the Calculus of Structures

There are two contributions in this paper. First, we give a direct proof of the known fact that Frege systems with substitution can be p-simulated by the calculus of structures extended with the substitution rule. This is done without referring to the p-equivalence of extended Frege systems and Frege systems with substitution. Second, we then show that the cut-free calculus of structures with substitution is p-equivalent to the cut-free calculus of structures with extension.