Law of Identity

Definition
Every proposition entails itself:
 * $$p \vdash p$$

From the modus ponendo ponens and the rule of implication, this is equivalent to:
 * $$\top \dashv \vdash p \implies p$$

A seemingly trivial rule, but can be surprisingly useful to get a particular formula into the right place in a proof.

The context can be expanded slightly:
 * $$p \dashv \vdash p$$

from which we immediately obtain:
 * $$\top \dashv \vdash p \iff p$$

Proof by Natural deduction
These are proved by the Tableau method.

This is the shortest tableau proof possible:

Proof by Truth Table
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.