Rule of Implication

Context
This is one of the axioms of natural deduction.

The rule
If, by making an assumption $$p$$, we can conclude $$q$$ as a consequence, we may infer $$p \Longrightarrow q$$:

$$\left({p \vdash q}\right) \vdash p \Longrightarrow q$$

This is sometimes known as:


 * "implies-introduction";
 * "conditional proof" (abbreviated CP).


 * Abbreviation: $$\Longrightarrow \mathcal{I}$$
 * Deduced from: The pooled assumptions of $$q$$.
 * Discharged assumption: The assumption of $$p$$.
 * Depends on: The series of lines from where the assumption of $$p$$ was made to where $$q$$ was deduced.

Explanation
This means: if we know that by making an assumption $$p$$ we can deduce $$q$$, then we can encapsulate this deduction into the compound statement $$p \Longrightarrow q$$.

Thus it provides a means of introducing a conditional into a sequent.