Validity of substituting identically distributed r.v. in expectation

Question

The question is motivated by increments of a standard brownian motion. Assuming $s<t$, I now $B_t-B_s$ is identically distributed from $B_{t+h}-B_{s+h}$. If I want to prove (in the trivial case, that $B_t$ is brownian motion) that $B_t-B_s$ is normally distributed, I use the characteristic function $E[e^{i \theta (B_t-B_s)}]$ and I must replace it, giving $E[e^{i \theta (B_{t-s}-B_0)}]=E[e^{i \theta(t-s)^{1/2}B_1}]$. At the end my question is if this is valid. I replaced $B_t-B_s$ for $B_{t-s}-B_0$ and then replaced it by $\sqrt{t-s}B_1$ all justified in assuming that if I replace a rv with another which is identically distributed, I would know E(f(X))=E(f(Y)).