prove this theorem $\vdash (\exists x_i (A\to B)\to (A\to \exists x_i B))$

Question

Here is my thought

to prove the theorem we should get $\{\exists x_i (A \to B), A\} \vdash \exists x_i B $

then, I don't know how to process...

Answer 1

Use $\exists{x_i}(A\rightarrow{B})\equiv\neg \forall{x_i}\neg(A\rightarrow{B})\equiv\neg \forall{x_i}(A\land\neg{B})$ and the instantiate universal quantifier.