Use conditional proof or indirect proof and the eighteen rules of inference to establish the truth of the following tautologies.

★1. P ⊃ [(P ⊃ Q) ⊃ Q]

2. (∼P ⊃ Q) ⋁ (P ⊃ R)

3. P ≡ [P ⋁ (Q P)]

★4. (P ⊃ Q) ⊃ [(P R) ⊃ (Q R)]

5. (P ⋁ ∼Q) ⊃ [(∼P ⋁ R) ⊃ (Q ⊃ R)]

6. P ≡ [P (Q ⋁ ∼Q)]

★7. (P ⊃ Q) ⋁ (∼Q ⊃ P)

8. (P ⊃ Q) ≡ [P ⊃ (P Q)]

9. [(P ⊃ Q) (P ⊃ R)] ⊃ [P ⊃ (Q R)]

★10. [∼(P ∼Q) ∼Q] ⊃ ∼P 11. (P ⊃ Q) ⋁ (Q ⊃ P)

12. [P ⊃ (Q ⊃ R)] ≡ [Q ⊃ (P ⊃ R)]

★13. (P ⊃ Q) ⊃ [(P ⊃ ∼Q) ⊃ ∼P]

14. [(P ⊃ Q) ⊃ R] ⊃ [(R ⊃ ∼R) ⊃ P]

15. (∼P ⋁ Q) ⊃ [(P ⋁ ∼Q) ⊃ (P ≡ Q)]

★16. ∼[(P ⊃ ∼P) (∼P ⊃ P)]

17. P ⊃ [(Q ∼Q) ⊃ R]
18. [(P Q) ⋁ R] ⊃ [(∼R ⋁ Q) ⊃ (P ⊃ Q)]
★19. P ≡ [P ⋁ (Q ∼Q)]
20. P ⊃ [Q ≡ (P ⊃ Q)]