Recently I have started reviewing mathematical notions, that I have always just accepted. Today it is one of the fundamental ones used in equations:
If we have an equation, then the equation holds if we do the same to both sides.
This seems perfectly obvious, but it must be stated as an axiom somewhere, presumably in formal logic(?). Only, I don’t know what it would be called, or indeed how to search for it – does anybody knw?
This axiom is known as the substitution property of equality. It states that if f is a function, and x=y, then f(x)=f(y). See, for example, Wikipedia.
For example, if your equation is 4x=2, then you can apply the function f(x)=x/2 to both sides, and the axiom tells you that f(4x)=f(2), or in other words, that 2x=1. You could then apply the axiom again (with the same function, even) to conclude that x=1/2.