Definition:Gentzen Proof System/Instance 1/Alpha-Rule

Definition
Let $\mathcal L$ be the language of propositional logic.

Let $\mathscr G$ be instance 1 of a Gentzen proof system.

The $\alpha$-rule allows for introducing an $\alpha$-formula in the conclusion.

Its precise formulations is as follows:

$(\alpha)$: For any $\alpha$-formula $\mathbf A$ and associated $\mathbf A_1, \mathbf A_2$ from the table of $\alpha$-formulas:


 * Given $U_1 \cup \left\{{\mathbf A_1}\right\}$ and $U_2 \cup \left\{{\mathbf A_2}\right\}$, one may infer $U_1 \cup U_2 \cup \left\{{\mathbf A}\right\}$.