Deductive Logic, Warren Goldfarb, Hackett Publishing, 2003, Part IV, Exercise 5(a), p. 285

We are asked to show, using the laws of identity, that (∃x)Fxa and (x)¬ Fxb together imply ¬ (a = b). In the original example, a and b are respectively y and z, but I have used the first letters of the alphabet to indicate clearly that they are constants rather than unbound variables. We can proceed in a number of ways. I use indirect proof.

1. (∃x)Fxa
2. (x)¬ Fxb
3. ∴¬ (a = b)
4. * ¬ ¬ (a = b) ......... AIP
5. * a = b ......... 4 DN
6. * (∃x)Fxb ......... 1,5 Id
7. * Fmb ......... 6 EI x/m
8. * ¬ Fmb ......... 2 UI x/m
9. * Fmb • ¬ Fmb ......... 7,8 Conj.
10. ¬ ¬ ¬ (a = b) ......... 4-9 IP
11. ¬ (a = b) ......... 10 DN