I was reading through Enderton's "Mathematical Introduction to Logic" and came across 2 examples, one a formula, one a relationship:
∃x(Qx→∀x Qx)
∀y∃x Pxy $\nvDash$ ∃x∀yPxy
Enderton claims the first one is a validity. I fail to see how this is the case. How would I show this?
For the second one, I explicitly defined a predicate P: a 2 place predicate that says ($y*x = 1)$ so the statement would read as following: For all y, there exists such that ($y*x = 1)$ but the right side would read: there exists x such that for all y, ($y*x = 1)$. Now if you do it the converse way, ie ∃x∀yPxy $\vDash$ ∀y∃x Pxy then this relationship is actually true. My question is, is the example I provided enough to show the relationship holds for (2)?
edit: Not an exact duplicate, since there is a second part of the question that is not answered in the linked thread.