Validity of □(P→ Q) → (◊P → ◊Q) in linear temporal logic
Think about what the formula means. The first part says that it's always true that $P$&nbsp;implies&nbsp;$Q$; the second part says that, if $P$&nbsp;is true somewhere, then $Q$&nbsp;must be true somewhere, too. Well, if $P$&nbsp;implies&nbsp;$Q$ then, if $P$&nbsp;is true somewhere, then $Q$&nbsp;had better be true in that place, too!