We prove lower bounds of the form $exp\left(n^{\epsilon_d}\right),$
$\epsilon_d>0,$ on the length of proofs of an explicit sequence of
tautologies, based on the Pigeonhole Principle, in proof systems
using formulas of depth $d,$ for any constant $d.$ This is the
largest lower bound for the strongest proof system, for which any
superpolynomial lower bounds are known.