Show that the following rules are always valid. Translate these rules into “natural language”.

8 (z → x) → ((z → y) → (z → (x ∧ y)));
9 ((x ∧ y) ∨ z) → ((x ∨ z) ∧ (y ∨ z)), ((x ∨ z) ∧ (y ∨ z)) → ((x ∧ y) ∨ z);
10 ((x ∨ y) ∧ z) → ((x ∧ z) ∨ (y ∧ z)), ((x ∧ z) ∨ (y ∧ z)) → ((x ∨ y) ∧ z);
11 (x → y) → (y → x);
12 x ∧ x → y;
13 y → (x ∨ x).