Show that if $\vDash\Phi \to \Psi$ then $\vDash\forall x.\Phi \to \forall x.\Psi$
I started by letting $I,\theta$ be arbitrary
Assume that $I,\theta\vDash\Phi \to \Psi$
Need to show that $I,\theta\vDash\forall x.\Phi \to \forall x.\Psi$
Assume that $I,\theta\vDash\forall x.\Phi$
(for all v $\in \Bbb D$ $I,\theta[x/y]\vDash\forall x.\Phi$)
Need to show that $I,\theta\vDash\forall x.\Psi$
(for all v $\in \Bbb D$ $I,\theta[x/y]\vDash\forall x.\Psi$)
Let v $\in \Bbb D$ be arbitrary,
$I,\theta\vDash\Phi \to \Psi$
This is where I got stuck. I don't know how to introduce the quantifier or x into this bit so I can proceed. Any help would be appreciated.