Definition:Beta-Formula/Table

Definition
From Classification of $\beta$-Formulas, we obtain the following table of $\beta$-formulas $\mathbf B$ and corresponding $\mathbf B_1$ and $\mathbf B_2$:


 * $\begin{array}{ccc}

\hline \mathbf B & \mathbf B_1 & \mathbf B_2\\ \hline

\neg \paren {\mathbf B_1 \land \mathbf B_2} & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \mathbf B_1 \lor \mathbf B_2 & \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \implies \mathbf B_2 & \neg \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \mathbin \uparrow \mathbf B_2 & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \neg \paren {\mathbf B_1 \mathbin \downarrow \mathbf B_2} & \mathbf B_1 & \mathbf B_2 \\ \neg \paren {\mathbf B_1 \iff \mathbf B_2} & \neg \paren {\mathbf B_1 \implies \mathbf B_2} & \neg \paren {\mathbf B_2 \implies \mathbf B_1} \\ \mathbf B_1 \oplus \mathbf B_2 & \neg \paren {\mathbf B_1 \implies \mathbf B_2} & \neg \paren {\mathbf B_2 \implies \mathbf B_1} \\

\hline \end{array}$