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This sequent is provable.
\begin{logicproof}{2}
\lnot p \imp q & \prem \\
\lnot \lnot \lnot q \land r & \prem \\
\lnot \lnot \lnot q & $\ande{1} 2$ \\
\lnot q & $\negnege 3$ \\
\lnot\lnot p & $\MT 1,4$ \\
p & $\negnege 5$ \\
p \land \lnot \lnot \lnot q & $\andi 6,3$
\end{logicproof}