We study Frege proofs for the one-to-one graph Pigeon Hole Principle defined on the n × n grid where n is odd. We are interested in the case where each formula in the proof is a depth d formula in the basis given by ∧, ∨, and ¬. We prove that in this situation the proof needs to be of size exponential in nΩ(1/d). If we restrict the size of each line in the proof to be of size M then the number of lines needed is exponential in n /(log M)O(d). The main technical component of the proofs is to design a new family of random restrictions and to prove the appropriate switching lemmas.