Name of the basic property of equalities that if a=b then f(a)=f(b).

Question

A basic, fundamental property of equalities is that, if one applies a function on both sides of an equality, the equality still holds. Formally: for any two objects $a$ and $b$ of type $T$ and a function $f(x)$ whose domain is $T$, if $a=b$ then $f(a) = f(b)$.

This is a property so basic and fundamental, that we use it unconsciously, automatically and without knowing its name. Wikipedia says it is called "Substitution property", but the property of replacing a term in an expression by its other side in a given equality is also called "substitution" 1 2.

So what is the name of the property related to applying a function in both sides of an equality (and not the name of the property related to replacing a term by its equal)?

Answer 1

what is the name of the property related to applying a function in both sides of an equality ?

The rule is the equality substitution axiom for functions :

$t = s → f[z/t] = f[z/s]$.

Consider the example : $x=3$.

We may derive from it the new equality $x+2=5$ using the above axiom with the function $f(z) := (z+2)$.

As you can see "applying a function in both sides of an equality " amounts exactly to "replacing a term by its equal" into a function.

Answer 2

Take a look at the definition of a function as a special type of set-theoretic relation. The property you describe is a direct consequence of this definition. I do not think, that the property has a particular name.

Answer 3

Leibniz's law

For more insight, see https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms.