Rule of Sequent Introduction

Context
Natural deduction.

Definition
Let the statement $$p$$ be a statement form in a proof.

Let $$p \vdash q$$ be a substitution instance of a sequent for which we already have a proof.

Then we may introduce, at any stage of a proof (citing "SI"), one of the following:


 * The conclusion $$q$$ of a sequent already proved, or
 * A substitution instance of such a conclusion, together with a reference to the sequent that is being cited.

This conclusion depend upon the pool of assumptions upon which $$p$$ rests.

Proof
Let the statement $$p$$ be a statement form in a proof.

Given $$p \vdash q$$, we may infer $$\vdash p \Longrightarrow q$$, from the Rule of Implication.

So we have now derived $$p \vdash q$$, and we also have $$p$$, so we may infer $$q$$ by Modus Ponendo Ponens.

Q.E.D.