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.