I'm a bit confused with the first-order statement which defines the strong induction in the answer to this question Strong induction and vacuous truth:
$(\forall n)[(\forall m)(m < n \rightarrow P(m)) \rightarrow P(n)] \rightarrow (\forall n)P(n)$
It made sense to me when I first looked at it, but then I played with some examples and realized that even if $P(m)$ was false, we would have that this statement $(\forall n)[\forall m(m < n \rightarrow P(m)) \rightarrow P(n)]$ would be true vacuously, since $ m < n \rightarrow P(m)$ would be false. Therefore as I understand since that is true, then we can conclude that $(\forall n)P(n)$.
I know for sure that I'm missing something, so that's why I'm writing this question - to figure out what I'm missing and why my conclusions are wrong.