Axiom:Leibniz's Law

Definition
Two objects $x$ and $y$ are equal if and only if every property $x$ has, $y$ has, and every property $y$ has, $x$ has.

Let $=$ represent the relation of equality and let $P$ be an arbitrary property. This relation can then be symbolically expressed as

$x = y \dashv \vdash P(x)\implies P(y)\land P(y)\implies P(x)$

For all $P$ in the Universe of Discourse.

Let $S$ be an arbitrary set. From Set Definition by Predicate this definition is equivalent to

$x = y \dashv \vdash x\in S \implies y\in S \land y\in S \implies x\in S$

For all $S$ in the Universe of Discourse.

It is important to note that though Gottfried Wilhelm von Leibniz himself used this law as the definition of equality, Alfred Tarski points out "to regard Leibniz's law here as a definition would make sense only if the meaning of the symbol "$=$" seemed to us less evident than that of [expressions such as $P(x) \iff P(y)$]". Leibniz's law can also be adapted as an axiom, or not adapted at all.