Prove $x-y=0 \iff x=y$ using only field axioms

Question

We have $x - y = 0$ and we need to prove $x = y$ using only field axioms. How to prove this?

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If I do $x-y=0 \implies x+y-y=y \implies x=y$ is this wrong?

Answer 1

$\Rightarrow$

$y=y+0=y+(x-y)=x$

$\Leftarrow$

$0=y+(-y)=x+(-y)=x-y$