Drawing Sound Conclusions from Unsound Premises

Given sets $\Phi1={\phi{11},...,\phi{1u(1)}}, ...,\Phi{z}={\phi{z1},...,\phi{zu(z)}}$ of boolean formulas, a formula $\omega$ follows from the conjunction $\bigwedge\Phii= \bigwedge \phi{ij}$ iff $\neg \omega\wedge \bigwedge{i=1}^z \Phii$ is unsatisfiable. Now assume that, given integers $0\leq ei < u(i)$, we must check if $\neg \omega\wedge \bigwedge{i=1}^z \Phi'i$ remains unsatisfiable, where $\Phi'i\subseteq \Phii$ is obtained by deleting $\,\,e{i}$ arbitrarily chosen formulas of $\Phii$, for each $i=1,...,z.$ Intuitively, does $\omega$ {\it stably} follow, after removing $ei$ random formulas from each $\Phii$? We construct a quadratic reduction of this problem to the consequence problem in infinite-valued \luk\ logic \L$\infty$. In this way we obtain a self-contained proof that the \L$_\infty$-consequence problem is coNP-complete.