Reference Request: Proof of $\mathrm{H}(\mathrm{Prim}\,\mathcal{H}) \cong \mathrm{Prim}\,\mathrm{H}(\mathcal{H})$ for cocommutative dg-Hopf algebras

Question

In Loday’s book Cyclic Homology the following theorem appears:

A.9 Theorem. On a cocommutative differential graded Hopf algebra $\mathcal{H}$ over a characteristic zero field $k$ the homology and primitive functors commute, $$ \operatorname{H}(\operatorname{Prim} \mathcal{H}) \cong \operatorname{Prim} \operatorname{H}(\mathcal{H}) \,. $$

For a proof of this theorem Loday refers to Appendix B of Quillen’s Rational Homotopy Theory, but it seems to me that the assertion does not appear there.

Question: Where can a proof of the above theorem can be found?