Rule of Association

Context
Natural deduction.

Definition
This rule is two-fold:

$$p \land \left({q \land r}\right) \dashv \vdash \left({p \land q}\right) \land r$$
 * Conjunction is associative:

$$p \lor \left({q \lor r}\right) \dashv \vdash \left({p \lor q}\right) \lor r$$
 * Disjunction is associative:

Its abbreviation in a tableau proof is $$\textrm{Assoc}$$.

Proof
These are proved by the tableau method:

$$p \land \left({q \land r}\right) \vdash \left({p \land q}\right) \land r$$:

$$\left({p \land q}\right) \land r \vdash p \land \left({q \land r}\right)$$

is proved similarly.

$$p \lor \left({q \lor r}\right) \vdash \left({p \lor q}\right) \lor r$$:

$$\left(p \lor q\right) \lor r \vdash p \lor \left(q \lor r\right)$$

is proved similarly.