What's the intuitive explanation of (∀xFx → Gn) ≡∃x(Fx →Gn)?

Question

Suppose there are m, n, o in the domain of x

Then

(1) (∀xFx → Gn) ≡ ((Fm∧Fn∧Fo)->Gn)

(2) ∃x(Fx →Gn) ≡ (Fm→Gn)∨(Fn→Gn)∨(Fo→Gn)

How can (1) and (2) be equivalent? Since (1) says that all Fm, Fn, Fo have to be true so that Gn. (2) say that it's possible that Fm is true will guarantee Gn.

Any explanation in English? I know how to prove it in predicate language, it's just that intuitively it feels strange.

Answer 1

Quoting your $(1)$ and $(2):$

$$∀xFx → Gn\tag{1a}$$ $$(Fm∧Fn∧Fo)→Gn\tag{1b}$$ $$∃x(Fx →Gn)\tag{2a}$$ $$(Fm→Gn)∨(Fn→Gn)∨(Fo→Gn)\tag{2b}$$

Let's consider the truth of $Gn.$

Now, $(2\text b)$ claims that one of its conditionals must be true. But since we don't know which, if we wish to guarantee $Gn$'s truth, we do require all three of $Fm,Fn,$ and $Fo$ to be true.

But this is precisely what $(1\text b)$ is claiming: if we wish to guarantee $Gn$'s truth, we need all three of $Fm,Fn,$ and $Fo$ to be true.

Does this give a better sense of how/why $(1)\equiv(2)?$