Leray-Hirsch theorem for cohomology

Statement
Let $$p:E \to B$$ be a fiber bundle with fiber space $$F$$, and $$R$$ be a commutative unital ring, such that the following hold:


 * $$H^n(F;R)$$ is a finitely generated free $$R$$-module for every $$n$$
 * There exist classes $$c_j \in H^{k_j}(E;R)$$ whose restrictions $$i^*(c_j)$$ form a basis for $$H^*(F;R)$$ in each fiber $$F$$, via the inclusion of the fiber in $$E$$ (the choice of these classes needs to be made independent of the fiber)

Then the map:

$$\Phi: H^*(B;R) \otimes_R H^*(F;R) \to H^*(E;R)$$

given by:

$$\sum_{jk} b_j \otimes i^*(c_k) \mapsto \sum_{jk} p^*(b_j) \smile c_j$$

is an isomorphism.