# Definition:Natural Deduction/Derived Rules

## Definition

In practically working with **natural deduction**, the following derived rules are useful.

### Rule of Substitution

Let $S$ be a sequent of propositional logic that has been proved.

Then we may infer any sequent $S'$ resulting from $S$ by substitutions for letters.

### Rule of Sequent Introduction

Let the statements $P_1, P_2, \ldots, P_n$ be conclusions in a proof, on various assumptions.

Let $P_1, P_2, \ldots, P_n \vdash Q$ be a sequent for which we already have a proof.

Then we may infer, at any stage of a proof (citing **SI**), the conclusion $Q$ of the sequent already proved.

This conclusion depends upon the pool of assumptions upon which $P_1, P_2, \ldots, P_n$ rest.

### Rule of Theorem Introduction

We may infer, at any stage of a proof (citing $\text {TI}$), a theorem already proved, together with a reference to the theorem that is being cited.

## Sources

This page may be the result of a refactoring operation.As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.In particular: Specifically the three rules mentioned here. Rest went to Definition:Rule of Inference and Definition:Natural Deduction/Rules of InferenceIf you have access to any of these works, then you are invited to review this list, and make any necessary corrections.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{SourceReview}}` from the code. |

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.15$: Rules of inference - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 1.2$: Conditionals and Negation - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs