+theorem eqb_true_to_eq: \forall n,m:nat.
+eqb n m = true \to n = m.
+intros.
+change with
+match true with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m].
+rewrite < H.
+apply eqb_to_Prop.
+qed.
+
+theorem eqb_false_to_not_eq: \forall n,m:nat.
+eqb n m = false \to n \neq m.
+intros.
+change with
+match false with
+[ true \Rightarrow n = m
+| false \Rightarrow n \neq m].
+rewrite < H.
+apply eqb_to_Prop.
+qed.
+