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I don't think that it would be necessary to simplify everything, in its current form one can derive the truth table of \small\tt P without too much work, \begin{array}{|c|c|c|c|c|c|c|c|}\hline \small\color{green}{p} &\small \color{green}{q} &\small \color{green}{r} &\small \lnot p &\small (q\oplus\lnot p) &\small \lnot(q\oplus\lnot p) &\small ((p\to \lnot(q \oplus \lnot p)) &\small \lnot r &\small (q\to p)&\small (\lnot r \oplus (q \to p))) & \small\tt P \\ \hline \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} \\ \hline \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} \\ \hline \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} \\ \hline \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} \\ \hline \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{#C00}{0} \\ \hline \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} \\ \hline \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} \\ \hline \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{royalblue}{1} & \small\color{#C00}{0} & \small\color{#C00}{0} \\ \hline \end{array}

https://math.stackexchange.com/questions/903952/find-the-kernel-of-linear-transformation

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0=10

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\text{false}

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