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, 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

 * Axiom:Axioms of Equality
 * Definition:Equals