Checking validity of propositional statement

Question

I was unable to solve this particular problem which asks to find the valid first order logic among below:

1. $\forall x(P(x)\to Q(x))\to (\forall xP(x)\to\forall xQ(x))$

2. $\exists x(P(x) \lor Q(x))\to (\exists xP(x) \to \exists xQ(x))$

3. $\exists x(P(x)\land Q(x)) \leftrightarrow (\exists xP(x)\land \exists xQ(x))$

4. $\forall x \exists y P(x,y) \to \exists y \forall xP(x,y)$

I interpret it as follows:

$P(x)$: $x$ is a princess

$Q(x)$: $x$ is a queen

1. $\forall x(P(x)\to Q(x))$: Anyone, who is a princess, is queen.

   $\forall xP(x)\to\forall xQ(x)$: If everyone is princess, then everyone is queen.

2. $\exists x(P(x) \lor Q(x))$: Someone is princess or queen (or both (as it is inclusive OR)).

   $(\exists xP(x) \to \exists xQ(x))$: If someone is princess then someone is queen.

3. $\exists x(P(x)\land Q(x))$: Someone is (both) princess and queen.

   $(\exists xP(x)\land \exists xQ(x))$: Someone is princess and someone (could be different one than the princess) is queen.

4. This I know for sure is incorrect:

   $\forall x \exists y P(x,y) \not \to \exists y \forall xP(x,y)$:

But I dont find any of these interpretations make any of four options valid. Am I making mistake?

Answer 1

The first one is correct. Keeping your example, if everyone who is a princess is also a queen, then if everybody is a princess then everybody is a queen ;)