$pW_t+\sqrt{1-p^2}\tilde{W_t}$ independent increments?

Question

I want to show if $W_t$ and $\tilde{W_t}$ are independent Brownian motions then $$X_t=pW_t+\sqrt{1-p^2}\tilde{W_t}$$ is also Brownian. I am stuck with independent increments. If $s<t\leq u<v$ then $W_v-W_u$ and $W_t-W_s$ are independent and $\tilde{W_v}-\tilde{W_u}$ and $\tilde{W_t}-\tilde{W_s}$ are independent. Can I then infer that $$X_v-X_u=p(W_v-W_u)+\sqrt{1-p^2}(\tilde{W_v}-\tilde{W_u})$$ and $$X_t-X_s=p(W_t-W_s)+\sqrt{1-p^2}(\tilde{W_t}-\tilde{W_s})$$ are independent?