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 substitution instance of a sequent for which we already have a proof.

Then we may introduce, at any stage of a proof (citing SI), either:
 * The conclusion $Q$ of the 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.

This is called the rule of sequent introduction.

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:
 * $\vdash P_1 \implies \paren {P_2 \implies \paren {P_3 \implies \paren {\ldots \implies \paren {P_n \implies Q} \ldots} } }$

Also known as
This rule is also known as the rule of replacement.

Also see

 * Rule of Theorem Introduction, which is a direct corollary of this.

Thus we can convert any sequent into a theorem so as to use results already calculated in order to prove further results.