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

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The discussion centers on the equivalence between the statements (∀v Fv -> p) and (∃u Fu -> p), with specific conditions regarding the free occurrences of variables. Participants seek clarification on the meaning of "occurring free at all" and the implications of variable u's occurrences in Fu. One suggested approach to prove the equivalence involves transforming the implications using logical identities, such as representing A -> B as ¬A ∨ B. The conversation highlights the need for a clearer understanding of the terms used in the equivalence. Overall, the equivalence remains a topic of interest, prompting requests for proof and further explanation.
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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
 
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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.
 

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