Let $E$ be a spectrum. Following the video linked below, we say that a spectrum $Z$ is $E$-acyclic if $E_*Z=0$ and a spectrum $X$ is $E$-local if $[Z,X]=0$ whenever $Z$ is $E$-acyclic.
According to the video, from these definitions it is possible to prove that if $X,Y$ are $E$-local, and if there is an map $f:X\to Y$ inducing isos on $E_*$, then $f$ induces isos on $\pi_*$. I messed around with this claim for a bit, but I could not see a way to make use of the locality assumption.
Video at specified time: https://youtu.be/ZsHtsx_A8j8?t=2313
Screenshot of the slide in the video: Lecture slide describing the above information and exercise.
Let $X \to Y$ be a $E$-equivalence of $E$-local objects. Let $F$ be its fiber. Since $X \to Y$ is a $E$-equivalence, $F$ is $E$-acyclic. Then, since $X$ and $Y$ are $E$-local, the fiber $F$ is $E$-local as well (use the five lemma). So $F \simeq 0$. Therefore, the map $X \to Y$ is a weak equivalence.