$(8)$ Show that the following propositions imply that there is a person that likes soccer and ping pong $($explain each step of your argu$-$ There is a person that likes playing video games and writing. $$ If a person likes dancing, then they do not like writing. $$ It a person likes playing video games, then they like playing ping pong. $$ Everyone likes dancing or soccer. Use the following predicate names in your formal argument. D$($x$)$: x likes dancing P$($x$)$: x likes playing ping pong S$($x$)$: x likes playing soccer V$($x$)$: x likes playing video games W$($x$)$: x likes writing