Rule of Simplification

Context
The rule of simplification is one of the axioms of natural deduction.

The rule
This is two axioms in one.


 * 1) If we can conclude $$p \and q$$, then we may infer $$p$$: $$ p \and q \vdash p$$
 * 2) If we can conclude $$p \and q$$, then we may infer $$q$$: $$ p \and q \vdash q$$

This is sometimes known as the rule of "and-elimination".


 * Abbreviation: $$\and \mathcal{E}_1$$ or $$\and \mathcal{E}_2$$
 * Deduced from: The pooled assumptions of $$p \and q$$.
 * Depends on: The line containing $$p \and q$$.

Explanation
Note that there are two axioms here in one. The first of the two tells us that, given a conjunction, we may infer the first of the conjuncts, while the second says that, given a conjunction, we may infer the second of the conjuncts.

At this stage, such attention to detail is important.