# Prove $\vDash (\forall x (\varphi \rightarrow \psi)) \rightarrow (\forall x \varphi \rightarrow \forall x \psi)$

Prove that all formulas of the following form are true in all structures: $$(\forall x (\varphi \rightarrow \psi)) \rightarrow (\forall x \varphi \rightarrow \forall x \psi)$$

Can someone give me a hint? I have thought about showing that this formula is a tautology in propositional logic which would give me the statement. Or using the substition axiom but I don't know how to proceed. (I also had to prove that the other direction of implication is not true in all structures where for which I constructed a structure as a counterexample.)

It is not a tautology of propositional logic since it is of the form $$A\to (B\to C).$$ You can either prove it formally and use soundness, or just argue informally about structures. I’ll opt for the latter since I don’t know what proof systems you use.
We need to show in an arbitrary structure, if we have $$\forall x (\varphi\to\psi)$$ and $$\forall x \varphi,$$ then we have $$\forall x\psi,$$ which means if $$a$$ is any element of the domain, $$\psi(a)$$ holds. Well, by our assumptions, we have $$\varphi(a)\to \psi(a)$$ and $$\varphi(a),$$ so it follows that $$\psi(a)$$ holds.