I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : How do I reach this goal? Which rules do I use and with which support steps to each rule (proofs to prove each step?) Using only inference rules, reit, quantifier rules. What have I missed writing out?
On line 9 referencing conjunction elimination from line 8 should derive either B(a) or K(a), but not R(a).
You might start this with the assumption on line 8 which is made to hopefully eliminate the existential in line 2.
Then use conjunction elimination to derive B(a) from that assumption. Use universal elimination from line 1 to derive B(a) → R(a). Those two lines allow one to derive R(a). Use conjunction elimination to get K(a) and then conjunction introduction to combine R(a) ∧ K(a). Use existential introduction to turn that into ∃x(R(x) ∧ K(x)).
That will allow you to discharge the assumption using existential elimination to complete the proof.
Here's the skeleton for the proof. You begin by assuming a witness to the existential in the premises, aiming to derive a witness for the existential in the conclusion.
| Ɐx (Bx → Rx) Premise |_ Ǝx (Bx ˄ Kx) Premise | |_ [c] Bc ˄ Kc Assumption | | : ˄E | | : ˄E | | : ⱯE | | : →E | | Kc ˄ Rc ˄I | | Ǝx (Kx ˄ Rx) ƎI | Ǝx (Kx ˄ Rx) ƎE