Axiom:Leibniz's Law

Axiom
Let $=$ represent the relation of equality and let $P$ be an arbitrary property.

Then:


 * $x = y \dashv \vdash P \left({x}\right) \iff P \left({y}\right)$

for all $P$ in the universe of discourse.

That is, two objects $x$ and $y$ are equal iff $x$ has every property $y$ has, and $y$ has every property $x$ has.

Application to Equality of Sets
Let $S$ be an arbitrary set.

From Set Definition by Predicate, the above formulation can be expressed as:


 * $x = y \dashv \vdash x \in S \iff y \in S$

for all $S$ in the universe of discourse.

This is therefore the justification behind the notion of the definition of set equality.

He used this law as the definition of equality.

However, Alfred Tarski notes:
 * 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 'every property $x$ has, $y$ has'].

Hence Leibniz's law can also be adopted as an axiom, or not adopted at all.

Also see

 * Axioms of Equality
 * Equals