Are these two statements(theorems) equivalent?

Question

I am given this theorem:

Let $H$ be a check matrix for a linear code $C$. Then $C$ has minimum distance $d$ iff. there exists a set of $d$, but no set of $d-1$, linearly dependent columns in $H$.

Is the following rephrasing accurate (identical and equivalent in meaning to the above)?

Let $H$ be a check matrix for a linear code $C$. Then the minimum distance of $C$ is $d \in \mathbb N$ such that there exists a set of $d$, but no set of $d-1$, linearly dependent columns in $H$.

Answer 1

Yes.

The only thing that it possibly differs in is that it includes the assertion that every linear code has a minimum distance. If $H$ is a finite matrix, this nuance makes no difference whatsoever.