Quantifier free formula equivalents

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Let $P_{i,j}$ be quantifier free first order formulas (finitely many), let $\forall P$ denote the universal closure of the formula $P$. The question is: is there a quantifier free formula $Q$ such that $\bigvee_j\bigwedge_i\forall P_{i,j}$ is logially equivalent to $\forall Q$?