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

We are asked to show that the two: (x)Ax and (x)(Ax • Ax) are equivalent. In other words, we must be able to go from one to the other and back.

1. * (x)Ax ......... ACP
2. * Ax ......... 1 UI x/x
3. * Ax ......... 2 Rep.
4. * Ax • Ax ......... 2,3 Conj.
5. * (x)(Ax • Ax) ......... 4 UG
6. (x)Ax ⊃ (x)(Ax • Ax) ......... 1-6 CP

1. * (x)(Ax • Ax) ......... ACP
2. * Ax • Ax ......... 1 UI x/x
3. * Ax ......... 2 Simp.
4. * (x)Ax ......... 3 UG
5. (x)(Ax • Ax) ⊃ (x)Ax ......... 1-4 CP