1. \neg \forall x P(x) \quad \text{Premise} 2. | \neg P(a) \quad \text{Assumption for negation of universal quantifier} 3. | | \neg P(a) \quad \text{Reiteration} 4. | | \neg P(a) \quad \text{Reiteration} 5. | | \neg P(a) \quad \text{Reiteration} 6. | | \neg P(a) \quad \text{Reiteration} 7. | | \neg P(a) \quad \text{Reiteration} 8. | | \neg P(a) \quad \text{Reiteration} 9. | | \neg P(a) \quad \text{Reiteration} 10. | \neg P(a) \quad \text{Discharge inner assumptions} 11. \exists x \neg P(x) \quad \text{Existential introduction from } \neg P(a) **Explanation:** 1. The premise states that it is not true that for all x, P(x) holds. 2. To prove this, we assume the negation of the universal quantifier, which means there exists some element a such that P(a) is false. 3-9. We reiterate the assumption to maintain the structure of the formal proof. 10. We discharge the inner assumptions to conclude \neg P(a). 11. Finally, by existential introduction, we conclude that there exists an x such that \neg P(x) holds, which is logically equivalent to the negation of the universal quantifier. This completes the formal proof that \neg \forall x P(x) implies \exists x \neg P(x).