\begin{definition}[Definition:Alpha-Formula/Table]
From Classification of $\alpha$-Formulas, we obtain the following table of $\alpha$-formulas $\mathbf A$ and corresponding $\mathbf A_1$ and $\mathbf A_2$:
::$\begin{array}{ccc}
\hline
\mathbf A & \mathbf A_1 & \mathbf A_2\\
\hline
\neg\neg \mathbf A_1 & \mathbf A_1 & \\
\mathbf A_1 \land \mathbf A_2 & \mathbf A_1 & \mathbf A_2 \\
\neg \left({\mathbf A_1 \lor \mathbf A_2}\right) & \neg \mathbf A_1 & \neg \mathbf A_2 \\
\neg \left({\mathbf A_1 \implies \mathbf A_2}\right) & \mathbf A_1 & \neg \mathbf A_2 \\
\neg \left({\mathbf A_1 \mathbin \uparrow \mathbf A_2}\right) & \mathbf A_1 & \mathbf A_2 \\
\mathbf A_1 \mathbin \downarrow \mathbf A_2 & \neg \mathbf A_1 & \neg \mathbf A_2 \\
\mathbf A_1 \iff \mathbf A_2 & \mathbf A_1 \implies \mathbf A_2 & \mathbf A_2 \implies \mathbf A_1 \\
\neg \left({\mathbf A_1 \oplus \mathbf A_2}\right) & \mathbf A_1 \implies \mathbf A_2 & \mathbf A_2 \implies \mathbf A_1 \\
\hline
\end{array}$
\end{definition}