Proof of the Schwinger-Dyson Equation in the operator formalism

Question

I am looking for a non-functional/non-path-integral proof of the Schwinger-Dyson equation in the operator formalism.

Specifically, I mean equation (7.12) in Quantum Field Theory and the Standard Model by Matthew Schwartz.

Schwartz proves the equation for the free case with $n =1$ between equations (7.5) and (7.10).

His proof uses:

1. the time ordering operator (for two fields)
2. the product rule
3. the derivative of a step function is a delta function
4. the commutation relations
5. the time ordering operator (again)

Unfortunately, he does not prove the general case, but merely states that "it is not hard to see that..."

Note: There is an explicit expression for the time ordering of $n$ scalar fields here: https://en.wikipedia.org/wiki/Path-ordering#Time_ordering

Any help would be greatly appreciated.