Condition to prove $f$ is a reduction A theorem says if $f$ is a computable function and we can prove $x \in A \Leftrightarrow f(x) \in B$, then we can use reduction so $A \leq_m B$.
But i'm confused if should I prove if :

$(x \in A \Rightarrow f(x) \in B )\land (f(x) \in B \Rightarrow x \in A)$

Or

$(x \in A \Rightarrow f(x) \in B ) \land(x \notin A \Rightarrow f(x) \notin B)$

in which circunstances because I've seen both in demonstration. Or maybe are they equivalent ?
For example, here is a solution to the empty string problem. But with $f$ such as, $f: \text{M accepts w} \to \text{M accepts } \epsilon \\ f(\langle M, \epsilon \rangle) = \langle M \rangle$, we can proove 1. Is it valid ?
It is called contraposition :
$f(x) \in B \Rightarrow x \in A$
is equivalent to
$x \notin A \Rightarrow f(x) \notin B$.
So, the two statements that you have seen are equivalent as well.