Rule of Sequent Introduction

Context
Natural deduction.

Definition
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 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 $$p_1, p_2, \ldots, p_n \vdash q$$ rests.

Proof
By hypothesis and substitution instance we have a proof, using primitive rules, of:

$$p_1, p_2, \ldots, p_n \vdash q$$

By the Extended Rule of Implication, we have:

$$\vdash p_1 \Longrightarrow \left({p_2 \Longrightarrow \left({p_3 \Longrightarrow \left({\ldots \Longrightarrow \left({p_n \Longrightarrow q}\right) \ldots }\right)}\right)}\right)$$

Q.E.D.

This means we can convert any sequent into a theorem, and use the Rule of Theorem Introduction, which is a direct corollary of this, to use results already calculated in order to prove further results.