Strong completeness for the predicate logic of the continuous t-norms

The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom $(\mathrm{\forall }x\phi {)}^2\leftrightarrow \mathrm{\forall }x{\phi }^2$ and the infinitary rule $$\frac{\phi \vee (\alpha \to {\beta }^n):n\in \mathbb{N}}{\phi \vee (\alpha \to \alpha \mathrm{\&}\beta )}$$ to achieve strong completeness with respect to continuous t-norms.