Prove: \(\neg(\neg Q \longrightarrow \neg P) \vdash P \wedge \neg Q\)

Prove: \(B \wedge \neg B \vdash C\)

Prove: \((A \longrightarrow B) \longrightarrow \neg B \vdash \neg B\)

Prove: \(F \vee G \longrightarrow H \wedge I \vdash \neg F \vee H\)

Prereqs for the following problem include: conditional intro, repeat rule

Prove: \(Q \vdash P \longrightarrow Q\)

Prereqs for the following problem include: conditional intro, principle of explosion, repeat rule

Prove: \(\neg P \vdash P \longrightarrow Q\)

Prereqs for the following problem include: conditional intro, principle of explosion, repeat rule

Prove: \(P \vee Q, \neg P \vdash Q\)

Prove:
$$\begin{align}
& A \longrightarrow C \longrightarrow B \\
& \neg C \longrightarrow \neg A \\
& A \\
\hline
& B
\end{align}$$

Prove:
$$\begin{align}
& M \longrightarrow \neg N \\
& M \\
& H \longrightarrow N \\
\hline
& \neg H
\end{align}$$

Prove:
$$\begin{align}
& R \vee S \longrightarrow T \longrightarrow K \\
& \neg K \\
& R \vee S \\
\hline
& \neg T
\end{align}$$

Prove:
$$\begin{align}
& A \vee B \\
& C \longrightarrow D \\
& A \longrightarrow C \\
& \neg D \\
\hline
& B
\end{align}$$

Prove:
$$\begin{align}
& \neg G \longrightarrow A \vee B \\
& \neg B \\
& A \longrightarrow D \\
& \neg G \\
\hline
& D
\end{align}$$

Prove:
$$\begin{align}
& R \vee S \longrightarrow T \\
& P \vee Q \longrightarrow T \\
& R \vee P \\
\hline
& T
\end{align}$$

Prove De Morgan's laws (1/2): \(\neg (P \vee Q) \dashv\vdash \neg P \wedge \neg Q\)

Prove De Morgan's laws (2/2): \(\neg (P \wedge Q) \dashv\vdash \neg P \vee \neg Q\)