Proofs III - Proving Theorems
Prove: (C ≡ ~D) ⊃ ~(C ≡ D)
| 1. | (C ≡ ~D) | assump,cp | sp1 | |
| 2. | ~~(C ≡ D) | assump, ip. | sp1, sp2 | |
| 3. | (C ≡ D) | DN | 2 | sp1, sp2 |
| 4. | (C ⊃ D) & (D ⊃ C) | BE | 3 | sp1, sp2 |
| 5. | (C ⊃ ~D) & (~D ⊃ C) | BE | 1 | sp1, sp2 |
| 6. | (C ⊃ D) | simp | 4 | sp1, sp2 |
| 7. | (D ⊃ C) | simp | 4 | sp1, sp2 |
| 8. | (C ⊃ ~D) | simp | 5 | sp1, sp2 |
| 9. | (~D ⊃ C) | simp | 5 | sp1, sp2 |
| 10. | (~D ⊃ D) | h.s. | 6,9 | sp1, sp2 |
| 11. | ~~D v D | CE | 10 | sp1, sp2 |
| 12. | D v D | DN | 11 | sp1, sp2 |
| 13. | D | rep | 12 | sp1, sp2 |
| 14. | (D ⊃ ~D) | h.s. | 7,8 | sp1, sp2 |
| 15. | ~D v ~D | CE | 14 | sp1, sp2 |
| 16. | ~D | rep | 15 | sp1, sp2 |
| 17. | D & ~D | conj | 13,16 | sp1, sp2 |
| 18. | ~(C ≡ D) | i.p. | 2-17 | sp1 |
| 19. | (C ≡ ~D) ⊃ ~(C ≡ D) | cp | 1-18 |