$[\![ \exists y \in \dot{x} \varphi(y) ]\!]$
$\displaystyle =\sup_{t}[\![ t \in \dot{x} \wedge \varphi(t) ]\!]$
$\displaystyle =\sup_{t}[\![ \varphi(t) ]\!] [\![ t \in \dot{x} ]\!]$
$\displaystyle =\sup_{t}[\![ \varphi(t) ]\!] \sup_{y \in \text{dom}(\dot{x})}[\![ y=t ]\!]\dot{x}(y)$
$\displaystyle =\sup_{y \in \text{dom}(\dot{x})}\dot{x}(y)\sup_{t}[\![ \varphi(t)\wedge y=t ]\!]$
$\displaystyle =\sup_{y \in \text{dom}(\dot{x})} \dot{x}(y)[\![ \varphi(y) ]\!]$