Modus Ponendo Ponens

Context
The modus ponendo ponens is one of the axioms of natural deduction.

The rule
If we can conclude $$p \implies q$$, and we can also conclude $$p$$, then we may infer $$q$$:


 * $$p \implies q, p \vdash q$$

This is also known as:


 * Modus ponens;
 * The rule of implies-elimination;
 * The rule of material detachment.


 * Abbreviation: $$\implies \mathcal{E}$$
 * Deduced from: The pooled assumptions of each of $$p \implies q$$ and $$p$$.
 * Depends on: Both of the lines containing $$p \implies q$$ and $$p$$.

Explanation
This means: if we know that $$p \implies q$$, and we also know $$p$$, then we also know $$q$$.

Thus it provides a means of eliminating a conditional from a sequent.