I would like to know if the following contraposition proof of the above mentioned equivalence is correct. Thanks in advance.
1 $$((\forall x: A(x)) \Rightarrow B) \Leftrightarrow (\exists x: ( A(x) \Rightarrow B) )$$
2 $$((\forall x: A(x)) \Rightarrow B) \Rightarrow (\exists x: ( A(x) \Rightarrow B) )$$ $$\land$$ $$(\exists : ( A(x) \Rightarrow B) ) \Rightarrow ((\forall x: A(x)) \Rightarrow B)$$
3 Let $$((\forall x: A(x)) \Rightarrow B)\equiv True$$ $$and$$ $$(\exists x: ( A(x) \Rightarrow B) )\equiv False$$
4 $$(\exists : ( A(x) \Rightarrow B) )\equiv False \Rightarrow A(x)\equiv True \land B\equiv False \Rightarrow \unicode{x21af} ((\forall x: A(x)) \Rightarrow B)\equiv False $$
5 therefore $$((\forall x: A(x)) \Rightarrow B) \Rightarrow (\exists x: ( A(x) \Rightarrow B) )$$
Now for the other way around:
6 Let $$((\forall x: A(x)) \Rightarrow B)\equiv False$$ $$and$$ $$(\exists x: ( A(x) \Rightarrow B) )\equiv True$$
7 $$((\forall x: A(x)) \Rightarrow B)\equiv False \Rightarrow A(x) \equiv True \land B \equiv False \Rightarrow \unicode{x21af} (\exists x: ( A(x) \Rightarrow B) )\equiv False$$
8 therefore $$(\exists x: ( A(x) \Rightarrow B) ) \Rightarrow ((\forall x: A(x)) \Rightarrow B)$$
My concern with this solution is that I incorrectly dealt with the quantifiers in 4th and 7th lines after respectively asserting the truth values of the subparts $A(x)$ and $B$. Apologies for the awkward formatting, I'm still getting the hang of it.