Apologies, I totally edited out that part of my comment after finding a much simpler proof. I think my new derivation is fine assuming that proof by truth table is valid (which should be uncontroversial given that he uses it soon after this to show that AND and NOT are an "adequate set" for representing every logic function).

Edit: I was not thinking clearly above. Of course proof by truth table is valid, because truth tables are the basis of the notion of logical equality, and Jayne's axioms don't make sense unless you accept that notion as a given.