Given a closed formula $B$ of a first order theory, if it's true in every countable model, is it true in every model? Given a closed formula $B$ of a first order theory, if it's true in every countable model, is it true in every model?
I'm not sure if this is true, but it sounds too powerful. 

Is the first order theory countable? If it is then this follows from the downward Lowenheim-Skolem theorem: if a countable first order theory has an infinite model it has a countable model.
Suppose that $B$ is false in a model of your theory, $T$. 
Either that model is countable, or it isn't. If it isn't then $T\cup \{\neg B\}$ has a countable model by the downward Lowenheim-Skolem theorem. Either way $T\cup \{\neg B\}$ has a countable model. So if if $B$ is true in all countable models of $T$ it's true in all models of $T$.
