Statement For $X$ and $Y$simply connectedfinite homotopy types and a morphism $f \colon X \longrightarrow Y$ between them, the following are equivalent:

$f$ is an $\mathbb{F}_p$-homology equivalence, i.e. $H_\ast(f,\mathbb{F}_p)$ is an equivalence;

$f$ is a $\mathbb{Z}_p$-homology equivalence, i.e. $H_\ast(f,\mathbb{Z}_p)$ is an equivalence;

$f$ is a $\mathbb{Z}_p$-weak homotopy equivalence, i.e. $\pi_\ast(f) \otimes \mathbb{Z}_p$ is an equivalence.