Need help with Fitch proof.

Question

I have trouble with fitch proofs. Con rules cannot be used.

First question:

Premises:

1.(¬P → (Q ∨ ¬R)) ∧ (¬R → ¬Q)

2. Q

3. Q → ¬P

Goal: R → ((Q ∨ ¬R) ∧ ¬P)

Second Question is without premise: the goal is

¬ (∃x Cube(x) ∧ ¬∃x Cube(x))

Third question:

Premises:

1. ∃x ((Pink(a) ∧ ∀y (Pink(y) → y = x)) ∧ Orange(x))

2. Pink(b) ∧ ∀y (Pink(y) → y = b)

Goal: Orange(b)

Any help would be greatly appreciated, thank you.

For the first question, I was thinking of starting with changing the goal into ¬R ∨ ((Q ∨ ¬R) ∧ ¬P) and for the second one, rephrasing the goal to ¬∃x Cube(x) ∨ ¬¬∃x Cube(x). For the third one I honestly have no idea...

Answer 1

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:

$\text O(b)$.