Rule of Implication

Axiom
The rule of implication is one of the axioms of natural deduction.

If, by making an assumption $p$, we can conclude $q$ as a consequence, we may infer $p \implies q$.

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

Tableau Form
In a tableau proof, the rule of implication can be invoked in the following manner:


 * 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.

Also known as
This is sometimes known as:


 * The rule of implies-introduction
 * Conditional proof (abbreviated $\text{CP}$).