In the book Introduction to Mathematical Logic of Samuel Buss, example III.$42$ says:
Assume that $\mathfrak{A}\not\vDash C[\sigma]$. (...) since $x$ is not free in $C$, $\mathfrak{A}\not\vDash C[\tau]$ for every $x$-variant $\tau$ of $\sigma$.
By contraposition, if $\mathfrak{A}\vDash C[\tau]$ for some $x$-variant $\tau$ of $\sigma$, then $\mathfrak{A}\vDash \exists x\, C[\sigma]$. However, I don't see how does this imply that $\mathfrak{A}\vDash C[\sigma]$ using that $x$ is not free in $C$. Can anybody guide me?