The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.5E, problem 3b, p. 562

Show that the sentence: (x)[Ax ⊃(Ax ⊃Bx)] ⊃(x)(Ax ⊃Bx), is a theorem of predicate logic.

1. (x)[Ax ⊃(Ax ⊃Bx)] ⊃(x)(Ax ⊃Bx)
2. * (x)[Ax ⊃(Ax ⊃Bx)] ... ACP
3. * * Ax ... ACP
4. * * Ax ⊃(Ax ⊃Bx) ... 2 UI x/x
5. * * Ax ⊃Bx ... 3,4 MP
6. * * Bx ... 3,5 MP
7. * Ax ⊃Bx ... 3-6 CP
8. * (x)(Ax ⊃Bx) ... 7UG
9. (x)[Ax ⊃(Ax ⊃Bx)] ⊃(x)(Ax ⊃Bx) ... 2-8CP