Need help regarding a proof in First Order Logic

First to show that the claim is false if $\phi$ is not a sentence but a formula containing free variables, we can take $ϕ(x,y):=\forall x(x^2+y^2=x^2-y^2)$ with $y$ as a free variable and with domain as the usual $\mathbb R$. So clearly $M, g \models ϕ$ under the assignment $g$ where $I_F^g(y)=0$, however, $M, g' \models ϕ$ doesn't hold under another assignment $g'$ where $I_F^{g'}(y)=1$. Second to show that the claim is true if $\phi$ is a sentence, consider now $ϕ(x,y):=\forall x \forall y((x+y)(x-y)=x^2-y^2)$. Then any variable assignment does not affect the truth value of sentence $\phi$, and obviously from elementary algebra we have $M, g \models ϕ$ under any $g$.