The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.6E, 3(d), p. 572

Show that the following is a theorem: (x)(y)(z)[(x = y • y = z)⊃ x = z]. The statement illustrates the transitive property of relations. The proof is quite straightforward:

(x)(y)(z)[(x = y • y = z)⊃ x = z]

1. * x = y • y = z ......... ACP
2. * x = y ......... 1 Simp.
3. * y = z ......... 1 Simp.
4. * x = z ......... 2,3 Id
5. (x = y • y = z)⊃ x = z ......... 1-4 CP
6. (z)[(x = y • y = z)⊃ x = z] ......... 5 UG
7. (y)(z)[(x = y • y = z)⊃ x = z] ......... 6 UG
8. (x)(y)(z)[(x = y • y = z)⊃ x = z] ......... 7 UG