# If $\mathcal{A}\models \Phi$, then for $\phi\in \Phi$, $\mathcal{A} \models \phi [\alpha]$ for all assignments $\alpha$?

If a structure $$\mathcal{A}$$ satisfies a set of formulas $$\Phi$$, then for $$\phi\in \Phi$$, does $$\mathcal{A} \models \phi [\alpha]$$ for all assignments $$\alpha$$?

I apologize if this question may be very vague. I'm just studying predicate logic right now.

• Unless I'm missing something, that's exactly what "$\mathcal{A}\models\phi$" means in the first place when $\phi$ has free variables. – Noah Schweber Nov 15 '18 at 0:44