Good evening! I have got a question about this following practice proof my professor has given me:
Premise:
- ∃x ((P(a) ∧ ∀y (P(y) → y = x)) ∧ R(x))
- P(b) ∧ ∀y (P(y) → y = b)
Prove: R(b)
I understand that this proof is true, however, I cannot seem to prove it using Fitch rules.
This is what I have so far. How should I go about proving this further? The way I was thinking was this: if I can somehow get to P(a) ∧ ∀y (P(y) → y = b)) ∧ R(b)), I can get R(b) through conjunction elimination. However, I am not sure how the second premise can even help with that.
Edit: It seems as if this proof can be done with only elimination rules. Still need to figure it out.
The key is you cannot substitute $b$ initially, only an arbitrary constant $c$ is allowed. Then you can try use your 2nd premise's 2nd conjunct to prove $c=b$ within the same subproof. Then you also have to invoke your related equality rules and $P(b)$ is needed here to let you invoke ∀-elim rule (let you safely substitute $b$ for $y$) in the subsequent subproof to arrive at $c=b$...