Rule of Implication

Context
The rule of implication 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 $$p \implies q$$:
 * $$\left({p \vdash q}\right) \vdash p \implies q$$

This is sometimes known as:


 * The rule of implies-introduction;
 * Conditional proof (abbreviated CP).

It can be written:
 * $${\begin{array}{|c|} \hline p \\ \vdots \\ q \\ \hline \end{array} \over p \implies q} \to_i$$


 * Abbreviation: $$\implies \mathcal I$$
 * 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 $$p \implies q$$.

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