I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\mathrm{d}t\tag{ 1}$$
to mean (formally)
$$\int_{t=s_0}^{t=s_1}\mathrm{d}X(t) = \int_{t=s_0}^{t=s_1}\sigma(t)\mathrm{d}W(t)+\int_{t=s_0}^{t=s_1}\alpha(t)\mathrm{d}t\text{.}\tag{2}$$
This is fine and makes sense. The problem is that he will often substitute the LHS of $(1)$ for the RHS of $(1)$ in other stochastic differential equations. For example, given the function $S(t)$, he may use $(1)$ to justify the equality
$$S(t) \mathrm{d}X(t) = S(t)\sigma(t)\mathrm{d}W(t)+S(t)\alpha(t)\mathrm{d}t\text{,}\tag{3}$$
which is equivalent to
$$\int_{t=s_0}^{t=s_1} S(t) \mathrm{d}X(t) = \int_{t=s_0}^{t=s_1} S(t)\sigma(t)\mathrm{d}W(t)+\int_{t=s_0}^{t=s_1} S(t)\alpha(t)\mathrm{d}t\text{.}\tag{4}$$
This trick appears to be purely symbolic. Intuitively, it makes sense. However, I am having trouble proving that you can make this substitution. Is there a proof that you can make these kinds of substitutions (in this example or in general)?