$\Sigma \vDash \exists x a(x)$ if and only if $\Sigma \vDash (\neg \forall x( \neg a(x))).$

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This is the proof I was provided in one direction.

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I am confused as to why there is a necessity to introduce variables $s, d$. How does one make the leap there $\exists d$. Any proof clarification would be really appreciated!