Can someone help me to prove this in S5 using Hilbert's axiomatic system?

Question

◊(P ∧ □Q) ⊢ ◊(P ∧ Q)

I wrote a modal tableaux and it was easy, but I cannot prove it using (H1-H3) axioms for propositional logic, S5-axioms, (MP) and (RN).

Answer 1

It is difficult to help without knowing explicitly what axioms you are working on. In any case, here are some general hints:

1. Recall that $\alpha \to \beta \vdash (\gamma \land \alpha) \to (\gamma \land \beta)$ is a theorem of propositional logic.
2. What happens if you set

   \begin{align} \alpha := Q \qquad\qquad \beta := \Box Q \qquad\qquad \gamma := P \end{align}

3. Also recall that $\Box (\phi \to \psi) \to (\Diamond\phi \to \Diamond\psi)$ is a $\mathsf{K}$-theorem.

Hope this helps!