The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 10.4E, 8(d), p. 558

Show that the following argument is valid.

1. (x)[(Fx • Gx) ≡ (∃y)(Axy • Py)]
2. (∃x)(∃y)(Fx • Axy • Py)
3. ∴(∃x)(Fx • Gx)
4. (∃y)(Fa • Aay • Py) ......... 2 EI x/a
5. Fa • Aam • Pm ......... 4 EI y/m
6. (Fa • Ga) ≡ (∃y)(Aay • Py) ......... 1 UI x/a
7. [(Fa • Ga) ⊃ (∃y)(Aay • Py)] • [(∃y)(Aay • Py) ⊃(Fa • Ga)] ......... 6 BE
8. Aam • Pm ......... 5 Simp.
9. (∃y)(Aay • Py) ......... 8 EG
10. (∃y)(Aay • Py) ⊃(Fa • Ga) ......... 7 Simp.
11. Fa • Ga ......... 9,10 MP
12. (∃x)(Fx • Gx) ......... 11 EG