This article presents a simplified proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving switching lemmas developed by Razborov and Beame. A similar result is also proved for some examples based on graphs.
"Simplified Lower Bounds for Propositional Proofs." Notre Dame J. Formal Logic 37 (4) 523 - 544, Fall 1996. https://doi.org/10.1305/ndjfl/1040046140