Rule of Implication

Proof Rule
The rule of implication is a valid deduction sequent in propositional logic: If, by making an assumption $p$, we can conclude $q$ as a consequence, we may infer $p \implies q$.

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

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
 * The rule of conditional proof (abbreviated $\text{CP}$).

Technical Note
When invoking Rule of Implication in a tableau proof, use the Implication template:



where:
 * is the number of the line on the tableau proof where the Rule of Implication is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof where the antecedent can be found
 * is the line of the tableau proof where the consequent can be found