$\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{Tot}{Tot}
need to show
\[\Hom(A,I)→\Tot^Π \Hom(P,I)←\Hom(P,B)\]
these maps are quasi-isomorphism.
$\Leftrightarrow$ cones are acylic.
Observation: $\operatorname{Cone}\left(\Hom(A,I)→\Tot^Π\Hom(P,I)\right)$ is the total complex of the double complex
$\Hom(P,I)$ with $\Hom(A,I)[-1]$ for this augmented double complex we see that $\Tot^Π$ is exact using the acyclic assembly lemma: $\Hom(P_p,-)$ is exact since $P_p$ is projective, $\Hom(-,I^q)$ is exact since $I^q$ is injective.
\[R^*\Hom(A,-)(B)=H^*\Hom(A,I)≅H^*\Tot^Π(\Hom(P,I))≅H^*\Hom(P,B)=R^*\Hom(-,B)(A)\]