 Truth-matrix countermodel for a Hilbert axiom set

Take ordinary material logical implication on $\{0,1\}$, designate $1$, and interpret negation by the constant-zero operation $N$. Then $A\to (B\to A)$ and $(NB\to A)\to (NA\to B)$ are always designated, and modus ponens rule and logical implication discharge preserve consequence. But $(NB\to NA)\to (A\to B)$ is false for $A=1,B=0$. Thus the first two schemes and the deduction rule do not derive the last scheme. This detects the loss of a genuine classical-negation principle even though all the displayed schemes are tautologies under the usual classical negation.