It seems your request is incomplete and a bit unclear. However, based on what I can understand, you might be asking for a theorem proof that involves symbols like $\forall$ (for all), $\exists$ (there exists), and possibly an "if and only if" (iff) statement. The theorem you're trying to prove may relate to logic or set theory.

Could you provide the full statement or clarify the theorem you're asking to prove? Here’s a general outline of how to prove a theorem involving such logic:

Template for Proving a Theorem Using Symbols:

1. Given Hypothesis:

   • Let $P(x)$ be a predicate or condition about a variable $x$.
   • You are asked to prove that some statement $Q$ holds if and only if (iff) $P(x)$.

2. Proof Structure for $P(x) \iff Q$:

   • Step 1: Prove $P(x) \implies Q$ (if $P(x)$, then $Q$).
     • Start by assuming