Definition:Deduction Rule

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

The Deduction Rule is the rule of inference:


 * From $U, \mathbf A \vdash \mathbf B$, one may infer $U \vdash \mathbf A \implies \mathbf B$

where:


 * $\mathbf A, \mathbf B$ are WFFs of propositional logic
 * $U$ is a set of WFFs

For a given proof system, the Deduction Rule can be either a basic rule of inference, or a derived rule.

Applicable Proof Systems
The Deduction Rule is either a rule of inference or a derived rule for the following proof systems:


 * Hilbert Proof System: Instance 1

This result is known as the Deduction Theorem.

Notation
In terms of sequents, the Deduction Rule can be denoted as:


 * $\dfrac{U, \mathbf A \vdash \mathbf B}{U \vdash \mathbf A \implies \mathbf B}$

Also see

 * Rule of Implication