Huh?? If you allow quantification over propositions, you are no longer using first order logic.

I think you were closer to being on track before your edit. The first thing to realize is that a fallacy is not a false statement. It is an invalid inference scheme or rule of inference.

So, with P and Q taken to be schematic variables (to be instantiated as propositions), the following is a fallacy (affirming the consequent):

P -> Q |- Q -> P

Or, you could have simply corrected the words "additional axiom" in the quoted claim to "additional axiom scheme".