Equivalence Relation Proof Writing

Question

Consider the following example:

let x,y $\in$ $\mathbb{R}$ and let $R$ be a relation on $\mathbb{R}$ such that $xRy$ if $|x|=|y|$. Prove that $R$ is an equivalence relation on $\mathbb{R}$.

Upon proving this statement, I was just curious if it was okay to approach it as an if-then statement? For example, when proving the symmetric property I approached it as follows in the first few lines of my proof:

If $|x|=|y|$ then $xRy$

as in, $|x|=|y|$ implies that $xRy$.

Answer 1

let x,y $\in$ $\mathbb{R}$ and let $R$ be a relation on $\mathbb{R}$ such that $xRy$ if $|x|=|y|$. Prove that $R$ is an equivalence relation on $\mathbb{R}$.

Upon proving this statement, I was just curious if it was okay to approach it as an if-then statement? ...

If $|x|=|y|$ then $xRy$

Yes, when it defines $R$ and says that:

$xRy$ if $|x|=|y|$

That is the exact same thing as:

If $|x|=|y|$ then $xRy$

That is, 'P if Q' is the same as 'If Q, then P'