My textbooks states the following equivalences without proof:
$$(\psi \to (\exists x)\phi(x)) \Leftrightarrow (\exists x)(\psi \to \phi(x))$$ $$(\psi \to (\forall x)\phi(x)) \Leftrightarrow (\forall x)(\psi \to \phi(x))$$
At first blush they seem plausible, but they lead to the following additional equivalences, and these really make no sense to me:
$$((\forall x)\phi(x) \to \psi) \Leftrightarrow (\exists x)(\phi(x) \to \psi))$$ $$((\exists x)\phi(x) \to \psi) \Leftrightarrow (\forall x)(\phi(x) \to \psi))$$
I have tried to find proofs for the first two equivalences, without success.
Can someone point me to a proof of the first two equivalences?
Thanks!
Edit: Clarification: as Arturo Magidin noted, $\psi$ is such that $x$ is not free in it.