  We study Frege proofs using depth-$d$ Boolean formulas for the Tseitin
contradiction on $n \times n$ grids. We prove that if each line in the proof is
of size $M$ then the number of lines is exponential in $n/(\log M)^{O(d)}$.
This strengthens a recent result of Pitassi et al. [PRT22]. The key technical
step is a multi-switching lemma extending the switching lemma of H\r{a}stad
[H\r{a}s20] for a space of restrictions related to the Tseitin contradiction.
The strengthened lemma also allows us to improve the lower bound for standard
proof size of bounded depth Frege refutations from exponential in $\tilde
\Omega (n^{1/59d})$ to exponential in $\tilde \Omega (n^{1/(2d-1)})$.
