# Rule of Sequent Introduction

## Theorem

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.

This is called the rule of sequent introduction.

## Proof

By hypothesis we have a proof of:

$P_1, P_2, \ldots, P_n \vdash Q$

Therefore we can include this proof in our current proof and arrive at $Q$ with the pool of assumptions upon which $P_1, P_2, \ldots, P_n$ rest.

$\blacksquare$

## Also known as

This rule is also known as the rule of replacement.

## Technical Note

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

{{SequentIntro|line|pool|statement|depends|sequent}}

where:

line is the number of the line on the tableau proof where Rule of Sequent Introduction is to be invoked
pool is the pool of assumptions (comma-separated list)
statement is the statement of logic that is to be displayed in the Formula column, without the $...$ delimiters
depends is the line (or lines) of the tableau proof upon which this line directly depends
sequent is the link to the sequent in question that will be displayed in the Notes column.