Using Fitch to proof ∀x Indiff(x,x). Help

2018-07-22 02:48:03

I am having a hard time solving this Fitch Proof.

Goal: ∀x Indiff(x,x)

I have to proof this goal using the following four premises: (might not need all of them)

P1: ∀x∀y(WeakPref(x,y)∨WeakPref(y,x))

P2: ∀x∀y∀z((WeakPref(x,y)∧WeakPref(y,z))→WeakPref(x,z))

P3: ∀x∀y(StrongPref(x,y)↔ ¬WeakPref(y,x))

P4: ∀x∀y(Indiff(x,y)↔(WeakPref(y,x)∧WeakPref(x,y)))

I was stuck with how to end up with a single variable.

Any suggestion is appreciated.

Thank you!