ETA: This exchange places Jeremy in an interesting position, doesn't it? By picking a fight with one of the active participants in the thread, he places himself in an awkward position in terms of moderating it. Let us see how this evolves, shall we?
No, Jeremy, it is not. There are some negations that have to be combed out, but ultimately, it's not even a syllogism. There are only two terms in play: Closed Minded People (C) and Progressive People (P).
The first statement, which I assert as true by definition, is "A member of C cannot be P" or more formally "There exist no Cs which are also Ps".
The conclusion is simply a logically equivalent re-arrangement of this "All Ps are not C".
That these are logically equivalent can be seen set theoretically, if you imagine a sort of Venn diagram, both statements merely assert that the big circle labelled C does not overlap with the big circle labelled P. Or you can apply the standard rules for distributing negations around in statements with universal and existential quantifiers, see also first order logic.
I attach the standard identity: