Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$

Question

If we wish to convert a statement of the form $\exists x \in A ~:~ P(x)$ into the form of an implication, would the correct conversion be

$$\exists x ~:~ x \in A \implies P(x)$$

Thanks

Answer 1

NO.

$(∃x ∈ A)P(x)$ is the abbreviation for: $∃x \ (x∈A ∧ P(x))$ and this one is not equivalent to $∃x \ (x∈A \to P(x))$.

$∃x \ (x∈A ∧ P(x))$ is false if there are no elements in $A$ (i.e. when $A$ is empty) while in this case $∃x \ (x∈A \to P(x))$ is true.