We study Frege proofs using depth-𝑑 Boolean formulas for the Tseitin contradiction on 𝑛 ×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, Ramakrishman, and Tan [2022 IEEE 62nd Annual Symposium on Foundations of Computer Science, 2022, pp. 445–456]. The key technical step is a multiswitching lemma extending the switching lemma of Håstad [J. ACM, 68 (2021), 1] 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 ˜Ω⁡(𝑛1/59⁢𝑑) to exponential in ˜Ω⁡(𝑛1/𝑑). This strengthens the bounds given in the preliminary version of this paper [J. Håstad and K. Risse, 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, 2022, pp. 1138–1149].