This sequent is not provable, counter example: \\
$\mathcal{M} : p = T, q = F, r = F$ \\
$\mathcal{M} \models (\lnot p \implies q) \land (q \implies r), \lnot r$ \\
$\mathcal{M} \nmodels \lnot \lnot \lnot r \land \lnot p$