why is this formula not valid for n=0 and i=0?

Question

$i$ and $n$ are integers.

$\exists (i)\Big(\big((n\geq 0) \wedge (0\leq i)\wedge (i\leq n)\big)\implies (3\leq n)\Big) $

It should be not valid for n=0 and i=0.

Answer 1

$i$ is a bound variable, so if you want to analyze it for a given value of $i$ you are ignoring the leading $\exists$. For $i=0, n=0$ all of the terms before the $\implies$ are true and the term after the $\implies$ is false, so the whole implication is false. If you consider the leading $\exists i$ the predicate is true for $n=0$ as $i=1$ is a witness. Both the antecedent and the consequent of the implication are false, which makes the implication true.