Suppose you have the following algebra $4= \{1,\ a,\ b,\ 0\}\ $ such that $0 \leq b \leq a \leq 1$. Now we define the model $V^{A}$ by transfinite recursion. Now consider the following sentence in the language of set theory:
$$\sigma =\exists\, u,v,w,x\ (x \in u \wedge x \notin v \wedge x \notin w \wedge u=v \wedge u=w )$$.
Is it possible that a single name say $\langle x, a\rangle$ is the witness for two existential quantifiers (say $\exists\, v$ and $\exists\, w$) ?