We study Frege proofs for the one-to-one graph Pigeon Hole Principle
defined on the $n\times n$ 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
$\land$, $\lor$, and $\neg$. We prove that in this situation the
proof needs to be of size exponential in $n^{\Omega (1/d)}$.
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.