Using the proof rules of predicate logic prove the validity of the following sequent:
$$ \forall X \exists Y(P(X)\lor Q(Y))\vdash \exists Y \forall X(P(X)\lor Q(Y)) $$
I have been trying to prove this for quite a long time but nothing at all seems to work.Any guidelines on how to go about it would really help. Thanks.
The Question is taken from Exercise 2.3 Q9(f) from the book Logic in computer science by Michael Huth and Mark Ryan.
I looked at your book and the system it uses seems to be the same as the Fitch system I like to use. The key to this proof is to set it up as a Proof by Contradiction: