MHB Is (∀v Fv -> p) Equivalent to (∃u Fu -> p)?

[agapito]

Consider the equivalence:

(∀v Fv -> p) <=> (∃u Fu -> p)

Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v.

Can someone please offer a proof of such equivalence. Many thanks. am

 

[Evgeny.Makarov]

I am not sure what it means to occur free "at all", and I don't understand the phrase "only those places that u occurs free in Fu". What is claimed about the places where u occurs free?

Perhaps your equivalence is $\forall v\,(Fv\to p)\iff (\exists u\,Fu)\to p$ or $(\forall v\,Fv)\to p\iff \exists u\,(Fu\to p)$. This is easy to show if you represent $A\to B$ as $\neg A\lor B$ and use de Morgan's law for quantifiers.