Consider a first-order language with ternary predicate P and equality predicate =. We define the formulas F1, F2, F3 as follows.

F1: ∀x∀y∃zP(x,y,z).

F2: ∀x∀y∀z∀u∀v∀w∀t (P(x,y,u) ∧ P(u,z,w) ∧ P(y,z,v) ∧ P(x,v,t) → =(w,t))

F3: ∀x∀y∀z∀w (∃u(P(x,y,u) ∧ P(u,z,w)) ↔ ∃v(P(y,z,v) ∧ P(x,v,w)))

a) Prove F1 ∧ F2 logically implies F3

b) Prove F2 does not logically imply F3

For B I got

Let the domain be {0,1}

Let P(x,y,z): x+y>z

Set x=0 y=1 u=0 z=1 w=0 v=0 t=0

Then F2 does not logically imply F3

Is this ok?

And how do I start with a) ?