Proof: is $ \forall x (P(x) \Leftrightarrow Q(x)) \Leftrightarrow (\forall x P(x) \Leftrightarrow \forall x Q(x))$

Question

I have to proof if the statement above is true or false. And How can i proof it appropriately?

Answer 1

It is not true.

E.g. on set $\{0,1,2\}$ let $P(1)$ be true and let $P(2),P(3)$ not be true.

Further let $Q(1),Q(2)$ be true and let $Q(3)$ not be true.

Then $\forall x\;P(x)\iff\forall x\;Q(x)$ is true but $\forall x\;(P(x)\iff Q(x))$ is not true.