The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.4E, 5(b), p. 557

Show that the set of sentences: (x)(Rx ≡ ¬ Rx) is inconsistent. The proof:

1. (x)(Rx ≡ ¬ Rx) ......... Assumption
2. Rx ≡ ¬ Rx ......... 1 UI x/x
3. (Rx ⊃ ¬ Rx) • (¬ Rx ⊃ Rx) ......... 2 BE
4. Rx ⊃ ¬ Rx ......... 3 Simp.
5. ¬ Rx ∨¬ Rx ......... 4 MI
6. ¬ Rx ......... 5 Taut.
7. ¬ Rx ⊃ Rx ......... 3 Simp.
8. ¬ ¬ Rx ∨ Rx ......... 7 MI
9. Rx ∨ Rx ......... 8 DN
10. Rx ......... 9 Taut.
11. Rx • ¬ Rx ......... 10,6 Conj.