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

The axiomatic system introduced by H\'ajek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom $(\forall x \phi)^2 \leftrightarrow \forall x \phi^2$ and the infinitary rule $$\frac{\phi \vee (\alpha \to \beta^n):n \in \mathbb{N}}{\phi \vee (\alpha \to \alpha \& \beta)}$$ to achieve strong completeness with respect to continuous t-norms.