equality

Equality is a binary relation between two quantities or expressions stating that they have the same value, or represent the same mathematical object, and we say they are equal.

Formally,
a relation $=$ is an equivalence relation if it is

• reflexive: $\mathrm{\forall }a$, then $a=a$.
• symmetric: $\mathrm{\forall }a,b$, if $a=b$ then $b=a$.
• transitive: $\mathrm{\forall }a,b,c$, if $a=b$ and $b=c$ then $a=c$.

and with respect to extensionality, we have Leibniz's Laws:

• if $x=y$, then any property of $x$ is a property of $y$
• if $x$ and $y$ have all properties in common, then $x=y$