Propositional logic/Zombie/Tautology/Exercise

This is how our artist thinks about a Zombie in a good mood.

We consider the following quotation of Sven Walter from the article "Zombies, Dualismus und Physikalismus“ (Zeitschrift for philosophische Forschung, Bd. 65, H. 2 (2011), pp. 241-254, https://www.jstor.org/stable/pdf/41346224.pdf).

" (${}P_{1}$) Zombies are imaginable.

(${}P_{2}$) If Zombies are imaginable, then Zombies are possible.

(${}P_{3}$) If Zombies are possible, then the physicalism is wrong.

Hence: the physicalism is wrong.“

Formalize the propositional rules used here, and show, with the help of truth tables, that they are tautologies.

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