Rule of Theorem Introduction

Definition

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.

Proof

This theorem is a corollary of the Rule of Sequent Introduction.

$\blacksquare$

Application

Using this rule, we can use theorems that we have derived in order to shorten proofs which may otherwise be long and unwieldy.

Technical Note

When invoking Rule of Theorem Introduction in a tableau proof, use the {{TheoremIntro}} template:

{{TheoremIntro|line|statement|theorem}}

where:

line is the number of the line on the tableau proof where the Rule of Theorem Introduction is to be invoked

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

theorem is the link to the theorem in question that will be displayed in the Notes column.

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 4$
• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules