(Proof Check): Homomorphism Theorem: $\phi(a)=\phi(b)$ if and only if $a\ker\ \phi=b\ker\ \phi$.

Question

In my textbook: Gallian's Contemporary Abstract Algebra, 9TH edition, on page 196, there is a homomorphism theorem that states the following:

Let $\phi$ be a homomorphism from G to G'.

Theorem: $\phi(a)=\phi(b)$ if and only if $a\ker\ \phi=b\ker\ \phi$.

I did my proof differently from the book and was wondering if it's possible to prove it this way:

$Proof$:

Since $\ker\phi=\{x\in G:\phi(x)=e\}$ we have:
$\phi(a)=\phi(b)$
$\phi(a)\phi(x)=\phi(b)\phi(x)$ (multiply both sides by $\phi(x)$
$\phi(ax)=\phi(bx)$ (property of homomorphism)
$a\ker\ \phi=b\ker\ \phi$. (This last part cannot be concluded, correct? Since $\phi$ is not a bijection.)

Answer 1

Hint:

$$a(\ker\phi)=b(\ker\phi)\iff ab^{-1}\in\ker\phi\iff\phi(ab^{-1})=e_{G'}$$

Now, use that the homomorphism properties of $\phi$ and conclude.