Questionnaire-2024/natural_deduction_propositional_logic/0024_sol.tex

\begin{logicproof}{2}
    p \implies q & \prem \\
    (p \land r) \lor q & \prem \\
    \begin{subproof}
        q \implies p & \assum \\
        \begin{subproof}
            p \land r & \assum \\
            r & $\ande{2} 4$ \\
            r \lor q & $\ori{1} 5$
        \end{subproof}
        \begin{subproof}
            q & \assum \\
            r \lor q & $\ori{2} 7$
        \end{subproof}
        r \lor q & $\ore 2,4-6,7-8$ \\
        \begin{subproof}
            p \land r & \assum \\
            p & $\ande{1} 10 $\\
            q & $\impe  1,11$ \\
            (s \land t) \lor q & $\ori{2} 12$
        \end{subproof}
        \begin{subproof}
            q & \assum \\
            (s \land t) \lor q & $\ori{2} 14$
        \end{subproof}
        (s \land t) \lor q & $\ore 2,10-13,14-15 $\\
        ((s \land t) \lor q) \land (r \lor q) & $\andi 16,9$
    \end{subproof}
    (q \implies p) \implies ((s \land t) \lor q) \land (r \lor q) & $\impi 3-17$
\end{logicproof}