Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(12), p. 435

The argument:

"Everything depends on something. All things that depend on everything are sustained by them. Everything that sustains all things depends on them. Therefore, there is something that sustains and is sustained by something."

1. (x)(∃y)Dxy

2. (x)(y)(Dxy ⊃Syx)

3. (x)(y)(Sxy ⊃Dxy)

4. ∴(∃x)(∃y)(Sxy • Syx)

5. (∃y)Dxy ......... 1 UI x/x

6. Dxa ......... 5 EI y/a

7. (y)(Dxy ⊃Syx) ......... 2 UI x/x

8. Dxa ⊃Sax ......... 7 UI y/a

9. Sax ......... 6,8 MP

10. (y)(Say ⊃Day) ......... 3 UI x/a

11. Sax ⊃Dax ......... 10 UI y/x

12. Dax ......... 9,11 MP

13. (y)(Day ⊃Sya) ......... 2 UI x/a

14. Dax ⊃Sxa ......... 13 UI y/x

15. Sxa ......... 12,14 MP

16. Sxa • Sax ......... 15,9 Conj.

17. (∃y)(Sxy • Syx) ......... 16 EG a/y

18. (∃x)(∃y)(Sxy • Syx) ......... 17 EG x/x