Translate the following inference to logic notation: If Thomas were a bartender, then Elisabeth is sleeping. For all x, if x_10 sleeps, then x_10 is a politician. If Elisabeth were a politician, then Thomas is a bartender. Therefore Elisabeth is sleeping if and only if Thomas is a bartender.
p -> P(a). forall x: P(x) -> Q(x). Q(a) -> p. Therefore P(a) <-> p.