Predicative logic implication, is it true?

Question

I'm struggling to find a counterexample for this expression...

$\left(\forall x P(x) \longrightarrow \forall x Q(X)\right) \overset{is\ this \ true??}{\longrightarrow} \forall x (P(x) \longrightarrow Q(x)) $

Answer 1

Let the universe be the set of integers and say

• $P(x)$ means "$x$ is even";
• $Q(x)$ means "$x$ is odd".

Then $$\forall x P(x) \longrightarrow \forall x Q(x)$$ is true, but $$\forall x (P(x) \longrightarrow Q(x))$$ is false.