It is not that these statements are "not generally valid"

The intended meaning of valid in my post is "valid step in a proof" in the given formal system. I reworded the offending section.

Obviously such statements will be true if H's axiom system is true, and in that sense they are always valid.

Yes, and one also has to be careful with the use of the word "true". There are models in which the axioms are true, but which contain counterexamples to Provable(#φ) → φ.