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

The task: show that the following argument is valid,

1. (x)(Px ⊃ Qx)
2. ∴ [(∃x)Px • (∃x)Qx)] ≡ (∃x)(Px • Qx)
3. * (∃x)Px • (∃x)Qx) ......... ACP
4. * (∃x)Px ......... 3 Simp.
5. * Pa ......... 4 EI x/a
6. * Pa ⊃ Qa ......... 1 UI x/a
7. * Qa ......... 5,6 MP
8. * Pa • Qa ......... 5,7 Conj.
9. * (∃x)(Px • Qx) ......... 8 EG
10. [(∃x)Px • (∃x)Qx)] ⊃ (∃x)(Px • Qx) ......... 3-9 CP
11. * (∃x)(Px • Qx) ......... ACP
12. * Pm • Qm ......... 11 EI x/m
13. * Pm ......... 12 Simp.
14. * (∃x)Px ......... 13 EG
15. * Qm ......... 12 Simp.
16. * (∃x)Qx ......... 15 EG
17. * (∃x)Px • (∃x)Qx ......... 14,16 Conj.
18. (∃x)(Px • Qx) ⊃[(∃x)Px • (∃x)Qx)] ......... 11-17 CP
19. {[(∃x)Px • (∃x)Qx)] ⊃ (∃x)(Px • Qx)} • {(∃x)(Px • Qx) ⊃[(∃x)Px • (∃x)Qx)]} ......... 10,18 Conj.
20. [(∃x)Px • (∃x)Qx)] ≡ (∃x)(Px • Qx) ......... 19 BE