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

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

It can be written:
 * $\displaystyle {p \land q \over p} \land_{e_1} \qquad \qquad {p \land q \over q} \land_{e_2}$


 * Abbreviation: $$\land \mathcal E_1$$ or $$\land \mathcal E_2$$
 * Deduced from: The pooled assumptions of $$p \land q$$.
 * Depends on: The line containing $$p \land q$$.

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
$$\begin{array}{|ccc||c|c|} \hline p & \land & q & p & q \\ \hline F & F & F & F & F \\ F & F & T & F & T \\ T & F & F & T & F \\ T & T & T & T & T \\ \hline \end{array}$$

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