I want to show that the definition of satisfiability is consistent with the definition given by relativization, i.e. Let $L=\{\in\}$. Let $M$ be a definable set and let $E\subset{M\times{M}}$. Let $\sigma(\bar{x})\in FM_{L}$. Let $\bar{a}\in{M^{lg{\bar{x}}}}$. I want to show that $\sigma^{M,E}(\bar{a}) \iff (M,E)\models \sigma(\bar{a})$. I can see that this should be done by induction on formulas but I'm having trouble seeing how. For example if $\sigma(\bar{a}):=a_{1}=a_{2}$ and suppose that $ZFC\vdash{a_{1}=a_{2}}$ but now $M$ may not interpret equality as set equality (or it might I'm not really sure). Also what about the converse, i.e if $M\models a_{1}=a_{2}$, then why does $ZFC\vdash{a_{1}=a_{2}}$?
$\sigma^{M,E}$ is defined using provability as found in standard texts like Jech or Kunen. This result appears in the third edition (pg 162) of Jech: "Let $\sigma$ be a formula of set theory. If M is a set and E is a binary relation on M and if $a_{1},...,a_{n}$ are elements of $M$, then $\sigma^{M,E}(\bar{a}) \iff (M,E)\models \sigma(\bar{a})$"