The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.4E, 11(b), p. 559

We are asked to show that the following sentences are inconsistent: {(∃x)(Hx • Mxc); (x)(Lx ⊃¬ Hx); (∃y)(Ly • Hy)}. In other words, we need to derive a contradiction. The thing to spot here is that we should start by instantiating the second of the existential statements (statement 3) and ignore the first. Otherwise we will not be able to instantiate statement 3 to the same constant as statement 2.

1. (∃x)(Hx • Mxc)
2. (x)(Lx ⊃¬ Hx)
3. (∃y)(Ly • Hy)
4. La • Ha ......... 3 EI y/a
5. La ......... 4 Simp.
6. Ha ......... 4 Simp.
7. La ⊃¬ Ha ......... 2 UI x/a
8. ¬ La ......... 6,7 MT
9. ¬ La • La ......... 8,5 Conj.