Deduction, Daniel Bonevac, Blackwell 2003, 7.3 problem 19

I have first established that the argument is valid by means of a truth tree. The deduction follows. On line 8 'z' gets instantiated by UI to 'x', which we are within the rules to do.

1. (x)(y)(z)[(Fxy • Fyz) ⊃Fxz]
2. ¬ (∃x)Fxx
3. ∴(x)(y)(Fxy ⊃¬ Fyx)
4. (x) ¬ Fxx / 2CQ
5. ¬ Fxx / 4UI
6. (y)(z)[(Fxy • Fyz) ⊃Fxz] / 1UI
7. (z)[(Fxy • Fyz) ⊃Fxz] / 6UI
8. (Fxy • Fyx) ⊃Fxx / 7UI
9. ¬ (Fxy • Fyx) / 5,8MT
10. ¬ Fxy ∨¬ Fyx / 9DeM
11. * Fxy / ACP
12. * ¬ Fyx / 11,10DS
13. Fxy ⊃¬ Fyx / 11-12CP
14. (y)(Fxy ⊃¬ Fyx) / 13UG
15. (x)(y)(Fxy ⊃¬ Fyx) / 14UG