The Poincaré-Miranda Theorem is an application of the cubical or polytope version of Sperner's lemma. For the proofs, one can consult the following interesting articles, by C. Ahlbach: https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1048&context=hmc_theses
and by M. Müger: https://arxiv.org/pdf/1310.8090.pdf
This last article also proves the theorem of invariance of dimension (two open sets in Euclidean spaces can only be homeomorphic if both dimensions agree) using Sperner's lemma.