I'll be delighted to get some help in the following:

Let D be the following system of inference: The axioms are all the tautologies except the ones that $\displaystyle \wedge$ is their main symbol and the only rule of inference if the rule:

$\displaystyle \frac{m,n}{m \wedge n} $ .

A. Prove that D is entire in the weak meaning, but not entire (I've proved it).

B. Let D' be the system of inference that derives from D by changing the rule by :

$\displaystyle \frac{m, m \to n}{n} $ . Prove that D' is entire.

C. Sketch a proof tree for : $\displaystyle P_1 \to (P_2 \vee P_3 ) $ from the set $\displaystyle \{P_1 , P_2, P_3 \}$ .

I've proved A... I realy need help in B and C.

Here is what I know:

Thanks !in part B: if we can prove somehow that D' is entire in the weak meaning, we will be able to deduce what we need (because of the special rule of inference we have) ... But how can we prove it?

In C-I know how to prove that $\displaystyle P_2 \vee P_3 $ is true in D'... But can't figure out how to prove the full statement...