Using natural deduction rules prove the statement:
$∀x∀y(P(x, y) → ¬P(y, x)),∀x∃yP(x, y)$ can be deduced to $¬∃v∀zP(z, v)$
I don't want to be given an answer, but a hint on how to start because I am stuck for a long time on this problem not knowing how to start efficiently.
Obviously here you can use proof by negation together with formal inference rules regarding quantifiers allowed in your system, and then a useful hint would be the predicate $P$ here can be interpreted as a strict partial order and thus $∃v∀zP(z,v)$ must be false when $z=v$, which is certainly an unavoidable special case for $∃v∀zP(z,v)$, while the given premise $∀x∃yP(x,y)$ can avoid such special case...