How to to prove that $\vdash p\rightarrow\Box\Diamond p$ in a system where $R$ is symmetric in non-classical logic?

Question

I am studying non-classical logic. I want to prove that $\vdash p\rightarrow\Box\Diamond p$ in a system where $R$ is symmetric

1. $\neg(p\rightarrow\Box\Diamond p),w_i$ from $R_{\neg}$
2. $p,w_i$ from $R_{\neg}$ on 1
3. $\neg \Box\Diamond p,w_i$ from $R_{\neg}$ on 1
4. $\Diamond\neg\Diamond p,w_i$ from $R_{\neg\Box}$ on 3
5. $w_iRw_j$ from $R_\Diamond$ on 4 ?

I didn't get this fifth step ...

Answer 1

It comes from $R\Diamond$ :

$$\underbrace{\Diamond A,wi}_{w_iRw_j\\A,w_j}$$

I should have written step 6 as well:

6. $\neg\Diamond p,w_j$ from $R\Diamond$ on 4

Then I could apply

7. $\Box\neg p,w_j$ from $R\neg\Diamond$ on 6
8. $w_jRw_i$ from $R_\Box$ on 7
9. $\neg p,w_i$ from $R_\Box$ on 7 and which leads to a contradiction with 1