Give a formal proof that uses rules of inferences to show th

Give a formal proof that uses rules of inferences to show that the given premises lead to the conclusion. Premises: Forall x(P(x) rightarrow S(x)) not x(Q(x)^R(x)) Forall x(not R(x) rightarrow not S(x)) Conclusion: Forall x(P(x) rightarrow not Q(x))

Solution

3) for all x (R(x) or !S(x)) or for all x (S(x) -> R(x))

2) for all x (!Q(x) or !R(x)) or for all x (R(x) -> !(Q(x))

1) for all x (P(x) -> S(x))

from 1,2 and 3 it is proved that

for all x (P(x) -> !Q(x))