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:
 * $$\top \dashv \vdash p \implies p$$

This is also known as the rule of repetition.

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:
 * $$\top \dashv \vdash p \iff p$$

This is also demonstrated in Equivalence Properties.

Proof by Natural deduction
By the tableau method:

This is the shortest tableau proof possible.

This is the second shortest tableau proof possible.

Proof by Truth Table
We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in both cases the truth value under the main connective is $$T$$ throughout for both models.

$$\begin{array}{|ccc||ccc|} \hline p & \implies & p & p & \iff & p \\ \hline F & T & F & F & T & F \\ T & T & T & T & T & T \\ \hline \end{array}$$

Interpretation by Models
Clearly, every model of $$P$$ is a model of $$P$$.

Thus by definition of logical consequence:
 * $$P \models P$$

Comment
Some sources, for example, use the statement:
 * $$\vdash p \implies p$$

to be the defining property of a tautology.