Difference between biconditional iff and equality

Question

I'm currently learning about mathematical proofs. I got confused about the concept of biconditional iff and equality "=". Suppose I want to prove c(a+b)=ca+cb. Is it correct to prove it by showing that c(a+b)$ \Leftrightarrow $ca+cb where I'll show c(a+b)$ \Rightarrow$ca+cb and ca+cb $ \Rightarrow$c(a+b)?

Answer 1

Equality is something that may or may not be true for a pair of mathematical expressions, like $a(b+c)$ and $ab + ac$. That pair happens to be equal when $a$, $b$ and $c$ are real numbers.

Equivalence is something that may or may not be true for a pair of logical statements, like "the quadrilateral $S$ is a square" and "the diagonals of $S$ are perpendicular and bisect each other".

The second kind of assertion (equivalence) is the kind you often prove by starting with each assertion as a hypothesis and proving the other.

The first kind you often prove with some kind of algebraic manipulation, or argument from the definitions. In case of the distributive law you would use the definitions of multiplication and addition.