$($a$)$ Define semantic entailment $\backslash$Gamma $\backslash$psi $,$ where $\backslash$Gamma be a $($possibly infinite$)$ set of

formulas in predicate logic and $\backslash$psi be a predicate logic formula.

$($b$)$ Let $\backslash$phi $1,\backslash$phi $2,$ and $\backslash$psi be three predicate logic formula. Let $\backslash$phi $1,\backslash$phi $2\backslash$psi $.$ Prove or disprove $\backslash$phi $1\backslash$psi $.$

$($c$)$ Let $\backslash$Gamma $1$ be a set of predicate logic formulas and $\backslash$phi and $\backslash$psi are also two predicate logic formulas. Prove that if $\backslash$Gamma $1,\backslash$phi $\backslash$psi and $\backslash$Gamma $1,\backslash$phi $,\backslash$psi $,$ then $\backslash$Gamma $1\backslash$psi $.$

$($d$)$ Prove using proof rules of natural deduction for predicate logic i$.$x$($P $($x$)->$Q$($x$))$x$($P $($x$)$ Q$($x$)).$

ii$.$x$($P$($x$)$Q$($x$))->$xP$($x$)$xQ$($x$).$