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

1 (x → x);

2 x → (y → x);

3 (x → y) → ((y → z) → (x → z));

4 (x → (y → z)) → ((x → y) → (x → z));

5 x → (x ∨ y), y → (x ∨ y);

6 (x → z) → ((y → z) → ((x ∨ y) → z));

7 (x ∧ y) → x, (x ∧ y) → y;