  We consider the G\"odel bi-modal logic determined by fuzzy Kripke models
where both the propositions and the accessibility relation are infinitely
valued over the standard G\"odel algebra [0,1] and prove strong completeness of
Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with
respect to this semantics. We axiomatize also the bi-modal analogues of $T,$
$S4,$ and $S5$ obtained by restricting to models over frames satisfying the
[0,1]-valued versions of the structural properties which characterize these
logics. As application of the completeness theorems we obtain a representation
theorem for bi-modal G\"odel algebras.
