Need help with Fitch proof.
Hint for the 3rd one
From 1st premise, instantiating it with $c$ new, we get:
$(\text P(a) ∧ ∀y (\text P(y) → y = c)) ∧ \text O(c)$,
from which: $\text P(a)$, $∀y (\text P(y) → y = c)$ and $\text O(c)$.
From 2nd premise we have $\text P(b)$ and using in with $∀y (\text P(y) → y = c)$, after instantiating it with $b$ we get:
$b=c$.
Finally, from $b=c$ and $\text O(c)$, using substitution axiom for equality we conclude with:
$\text O(b)$.