Identifying whether two statements are same. The two statements are:
$\forall a\in A, \exists b\in B, M(a)\implies N(b)$
$(\forall a\in A, M(a))\implies (\exists b\in B, N(b))$
I am not so good in logics. I feel the second statement implies first one. The existence of $b$ is not depending on $a$. So, I think two statements are same.
Am I correct? Can anyone give some counter for sets and predicates if No. Thank you.

They are not the same. Consider the following statements:

*

*$M(a)$: Person $a$ has a dog

*$N(b)$: Person $b$ has a cat

Then the first statement says that for any person, if that person has a cat, then there is a person with a dog. In other words, if there is a cat, then there is a dog.
The second sentence says that if every person has a cat, then someone has a dog.
They are seen to be inequivalent if some but not all people have cats, and no one has a dog.
