Given the two first order formulas:
$$ \vartheta = \exists y \forall x p(x,y) \\ \psi = \forall x \exists yp(x,y) $$
Show with FOL semantics that the following formula ia a valid first order formula.
$$ \vartheta \rightarrow \psi $$
Any idea on how to approach this?
Hint: Let $M$ be an $L$-structure, where $L$ is our language. Argue that if $\vartheta$ is true in $M$, then $\varphi$ is true in $M$.
We start the process like this. Because $\vartheta$ is true in $M$, there is an $a$ in the underlying set of $M$ such that for any $b$ in the underlying set of $M$, if $p_M$ is the interpretation of $p$ in $M$, then $p_M(b,a)$ is true in $M$.