Desired Conclusion: ∃𝑥(𝐹(𝑥)→𝐺(𝑥))
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Hint 1: The claim is clearly only valid in nonempty domains (you cannot prove something exists where nothing exists), so there is an unwritten assertion that something does exist.
Hint 2: You may syntactically prove by Excluding the Middle. Either there does exists something satisfying F (ie
∃x F(x)), or there does not.
Here's a skeleton using Fitch notation for Natural Deduction.
| ∃x x=x existential import |_ ∃x F(x) → ∃x G(x) premise | ∃x F(x) v ~∃x F(x) law of excluded middle | |_ ∃x F(x) assumption | | : | | ∃x (F(x)→G(x)) | + | |_ ~∃x F(x) assumption | | : | | ∃x (F(x)→G(x)) | ∃x (F(x)→G(x)) disjunction elimination
You won't be able to prove it, since these are not equivalent.
The premise says if there exists an x such that F(x) is true then there exists an x (not necessarily the same x) such that G(x) is true.
In the desired conclusion it is saying that there exists an x such that if F(x) is true then G(x) is true with that same x.