QUESTION $9$

$[35]$

Using the rules of natural deduction, prove the validity of the following sequents in predicate logic. In all cases, number your steps, indicate which rule you are using and indicate subproof boxes clearly.

Question $9\mathrm{.}1$

EExF$(x)vvEExG(x)|--$EEx$(F(x)vvG(x))||$

Question $9\mathrm{.}2$

AAx$(P(x)vvQ(x)),$AAx$(notP(x))|--$AAxQ$(x)||$

Question $9\mathrm{.}3$

AAxAAyAAz$((S(x,y{)}^{{?}^{?}}S(y,z))S(x,z)),$AAxnotS$(x,x)|--$AAxAAy$(S(x,y)$notS$(y,x))||$

Question $9\mathrm{.}4$

AAxAAy$(Q(y)F(x))|--$EEyQ$(y)$AAxF$(x)||$

Question $9\mathrm{.}5$

AAx$(P(x{)}^{{?}^{?}}Q(x))(x)|--$AAx$(PQ(x))||$