Let's say $M$ and $N$ are structures and $s$ is a variable assignment for $M$. Let's also say that $M$ is a substructure of $N$. If $\varphi$ is a formula then I need to prove that if $M,s \vDash \exists x\varphi$ then $N,s \vDash \exists x \varphi$. I can appeal to the fact that $M,s \vDash \varphi$ iff $N,s \vDash \varphi$. $\varphi$ also needs to be quantifier free in order to use this fact. To start off, I know that if $M,s \vDash \exists x \varphi$ then there exists some $x$-variant $s'$ of $s$ such that $M,s' \vDash \varphi$. From there I can use the above fact to say that if $M,s' \vDash \varphi$ then $N,s' \vDash \varphi$.
I'm confused where to go from here. I need to say that $N,s \vDash \exists x \varphi$ but I'm not sure how to get there from $N,s' \vDash \varphi$.