The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill, 2004, 4th edition, 10.6, ex.4(d)

We are asked to symbolize the argument and prove its validity. The argument is:

Rebecca loves those and only those who love her. The brother of Charlie loves Rebecca. Sam is Charlie's brother. So Sam and Rebecca love each other.

1. (x)(Lxr ≡ Lrx)
2. (∃x)[Bxc • (y)(Byc ⊃y = x) • Lxr)
3. Bsc
4. ∴ Lsr • Lrs
5. * ¬ (Lsr • Lrs) / AIP
6. * ¬ Lsr ∨¬ Lrs / 5DeM
7. * Lsr ≡ Lrs / 1UI x/s
8. * (Lsr ⊃Lrs) • (Lrs ⊃Lsr) / 7BE
9. * Lsr ⊃¬ Lrs / 6MI
10. * Lrs ⊃Lsr / 8Simp.
11. * Lrs ⊃¬ Lrs / 9,10HS
12. * ¬ Lrs ∨¬ Lrs / 11MI
13. * ¬ Lrs / 12Taut.
14. * Bac • (y)(Byc ⊃y = a) • Lar / 2EI x/a
15. * (y)(Byc ⊃y = a) / 14Simp.
16. * Bsc ⊃s = a / 15UI y/s
17. * s = a / 3,16MP
18. * Lar / 14Simp.
19. * Lsr / 17,18Id
20. * Lsr ⊃Lrs / 8Simp.
21. * Lrs / 19,20MP
22. * Lrs • ¬ Lrs / 21,13Conj.
23. ¬ ¬ (Lsr • Lrs) / 5-22IP
24. Lsr • Lrs / 23DN