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

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

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

The $\beta$-rule allows for introducing a $\beta$-formula in the conclusion.

Its precise formulations is as follows:

$(\beta)$: For any $\beta$-formula $\mathbf B$ and associated $\mathbf B_1, \mathbf B_2$ from the table of $\beta$-formulas:


 * Given $U_1 \cup \left\{{\mathbf B_1, \mathbf B_2}\right\}$, one may infer $U_1 \cup \left\{{\mathbf B}\right\}$.