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.

It can be written:
 * $${p \and q \over p} \and_{e_1} \qquad \qquad {p \and q \over q} \and_{e_2}$$


 * 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.

Demonstration by Truth Table
$$\begin{array}{|ccc||c|c|} \hline p & \and & q & p & q \\ \hline F & F & F & F & F \\ F & F & T & F & T \\ T & F & F & T & F \\ T & T & T & T & T \\ \hline \end{array}$$

As can be seen, when $$p \and q$$ is true so are both $$p$$ and $$q$$.