Apologies if this question is already posted.As it is hard to search $\LaTeX$ in google and first order logic in nothing without $\LaTeX$ .But i am sure i have different doubt than that of already posted one!
Question
Check if the given formula is valid or not.
$1.\forall x \exists y P(x,y) \Rightarrow \exists y \forall x P(x,y)$
$2.\forall x (P(x) \Rightarrow Q(x))\Rightarrow \left ( \forall x P(x) \Rightarrow \forall x Q(x) \right )$
My Approach
$1.\forall x \exists y P(x,y) \Rightarrow \exists y \forall x P(x,y)$
$x=\text{set of all boys}$
$y=\text{set of all girls}$
$P(x,y)=\text{x loves y}$
LHS=All boys loves some girls
RHS=some girl is loved by every boy. Hence $LHS \rightarrow RHS $ is not valid as LHS=true and RHS=false
$2.\forall x (P(x) \Rightarrow Q(x))\Rightarrow \left ( \forall x P(x) \Rightarrow \forall x Q(x) \right )$
$x=\text{set of natural numbers}$
$P(x)=x \text{which is multiple of }4$
$Q(x)= \text{Even natural numbers }$
LHS= $\forall x (P(x) \Rightarrow Q(x))$ which is always true.
RHS=$\left ( \forall x P(x) \Rightarrow \forall x Q(x) \right )$
which says that if everything is multiple of $4$ then everything is even which is not true.
Hence $LHS \rightarrow RHS $ is not valid as LHS=true and RHS=false
But i feel that $2$ is incorrect. Please help!