Let $R\subset A^2$ be an equivalence relation. Prove that $b \in [a] \implies [a]=[b]$.
$\textbf{Proof}$
Suppose that $b \in [a]$. By definition of equivalence class our hypothesis is equivalent to $bRa$, and by the symmetric property of $R$ we have that $bRa$ if and only if $aRb$, but by the transitive property of $R$ we have that $bRa$ and $aRb$ is the case if and only if $bRb$. By definition of equivalence class we know that $bRb$ is equivalent to $b \in [b]$. Therefore, $b \in [a]$ if and only if $b\in [b]$ and by the axiom of extensionality we conclude that $[a]=[b]$. Q.E.D.
I want to know if my proof is correct, can anyone help me please? Thank you so much
You probably mean a true thing, but "$bRa$ and $aRb$ is the case if and only if $bRb$" is false as stated. By reflexivity, $bRb$ is always true, so there's no need to find a proof for that.
More seriously, your use of the axiom of extensionality is invalid. If you had proved that for all $c$ it is true that $c\in[a]$ if and only if $c\in[b]$, then that would be a correct use; but you only have $b\in[a]$ if and only if $b\in[b]$ for the one specific element $b$.