This doesn’t work, for example if all three systems have tautology (1) as an axiom, let b=c=1 and let a be a singleton. Then at least all axioms of A become axioms in your system, which clearly shouldn’t happen for E
No. For each axiom a of A, “a or tautology” becomes an axiom. But “x or tautology” is itself a tautology for any x, so this isn’t a problem; “a or tautology” does not imply “a”.
No. For each axiom a of A, “a or tautology” becomes an axiom. But “x or tautology” is itself a tautology for any x, so this isn’t a problem; “a or tautology” does not imply “a”.
That’s true. Sorry for not being careful, everything checks out. Something seems to have short-circuited in my mind.