Advanced Math Help with Any Problems!

Show that the argument form with premises ${\displaystyle (p\wedge t)\Rightarrow (r\vee s),q\Rightarrow (u\wedge t),u\Rightarrow p,}$ and ${\displaystyle urc\text{or}\ne rs}$ and conclusion ${\displaystyle q\Rightarrow r}$ is valid by using rules of inferences.
$\begin{array}{|ccc|}\hline \text{Rule of Inference}& \text{Tautology (Deduction Theorem)}& \text{Name}\\ P& P\Rightarrow (P\vee Q)& \text{Addition}\\ \overline{\therefore P\vee Q}& & \\ P\wedge Q& (P\wedge Q)\Rightarrow P& \text{Simplification}\\ \therefore P& & \\ P& \left[\right(P)\wedge (Q\left)\right]\Rightarrow (P\wedge Q)& \text{Conjunction}\\ Q& & \\ \therefore \overline{P\wedge Q}& & \\ P& \left[\right(P)\wedge (P\Rightarrow Q\left)\right]\Rightarrow (P\wedge Q)& \text{Modus Ponens}\\ P\Rightarrow Q& & \\ \overline{\therefore Q}& & \\ ⌝Q& \left[\right(⌝Q)\wedge (P\Rightarrow Q\left)\right]\Rightarrow ⌝P& \text{Modus Tollens}\\ P\Rightarrow Q& & \\ \overline{\therefore ⌝P}& & \\ P\Rightarrow Q& \left[\right(P\Rightarrow Q)\wedge (Q\Rightarrow R\left)\right]\Rightarrow (P\Rightarrow R)& \text{Hypothetical Syllogism (''chaining'')}\\ Q\Rightarrow R& & \\ \overline{\therefore \Rightarrow R}& & \\ P\vee Q& \left[\right(P\vee Q)\wedge (⌝P\left)\right]\Rightarrow Q& \text{Disjunctive syllogism}\\ ⌝P& & \\ \overline{\therefore Q}& & \\ P\vee Q& \left[\right(P\vee Q)\wedge (⌝P\vee R\left)\right]\Rightarrow (Q\vee R)& \text{Resolution}\\ ⌝P\vee R& & \\ \overline{\therefore Q\vee R}& & \\ \hline\end{array}$

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{1,2,3,4,6,8,12}

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