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Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathrm{𝒜ℱ}$ modules. The main result is that if the $R$-module ${\mathrm{Ext}}_R^t(R/𝔞,M)$ is finite (finitely generated), $H_{𝔞}^i\left(M\right)$ is $𝔞$-cofinite for all $i<t$ and $H_{𝔞}^t\left(M\right)$ is minimax then $H_{𝔞}^t\left(M\right)$ is $𝔞$-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H_{𝔞}^i\left(N\right)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology module...