I know that ∀x∈A P(x) is only equivalent to ∀x(x∈A -> P(x)), but why isn't it equivalent to ∀x(x∈A /\ P(x))? I can use the laws in logic to compare the two, but I want an intuitive reason as to why this is the case. To me, it seems logical that if ∃x∈A P(x) equivalent to ∃x(x∈A /\ P(x)), then ∀x∈A P(x) is equivalent to ∀x(x∈A /\ P(x)), but I'm definitely wrong, so I'm asking for a new way to see it.
∃x∈A P(x) - "There exists an X in A that is P(X)".
∀x∈A P(x) - "All X in A are P(X)".
In the former statement, you use the "in A" part to provide more information about some specific X. This is expressed as a predicate (x∈A in this case) which is conjuncted with the other predicates representing the rest of your knowledge.
In the latter, you use the "in A" part to provide a condition under which P(X) holds, which is expressed as an implication.
Or, in other words - if there exists a cat that is black, there exists something that is a cat and is black. But if all cats are black, you can't say everything in the whole world is a cat and is black - but you can say that everything is black if it's a cat.