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OP's equation is a special case of

$$\left< \Omega \left| T\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(y)}\right\}\right| \Omega \right>_J~=~0, \tag{A}$$ where the Euler-Lagrange expression is $\frac{\delta S[\phi;J]}{\delta \phi^{\alpha}(x)}.$

NB: Be aware that the time-ordering usually used in the literature is the covariant time-ordering $T_{\rm cov}$, i.e. time-differentiations inside its argument should be taken after/outside the usual time ordering $T$. This induces quantum corrections/contact terms, so that eq. (A) can be formally rewritten into the standard Schwinger-Dyson equations:

$$ \left< \Omega \left| T_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(y)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| T_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(y)} \right\}\right| \Omega \right>_J ~. \tag{B}$$ One should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this & this Phys.SE posts.

OP's equation is a special case of

$$\left< \Omega \left| T\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~0, \tag{A}$$ where the Euler-Lagrange expression is $\frac{\delta S[\phi;J]}{\delta \phi^{\alpha}(x)}.$

NB: Be aware that the time-ordering usually used in the literature is the covariant time-ordering $T_{\rm cov}$, i.e. time-differentiations inside its argument should be taken after/outside the usual time ordering $T$. This induces quantum corrections/contact terms, so that eq. (A) can be formally rewritten into the standard Schwinger-Dyson equations:

$$ \left< \Omega \left| T_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| T_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(x)} \right\}\right| \Omega \right>_J ~. \tag{B}$$ One should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this & this Phys.SE posts.

First of all, OP's eq. (3) needs a quantum correction given by the Schwinger-Dyson equation:

$$\tag{A} \left< T\left\{ F[\varphi]\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}\right\}\right>~=~i\hbar\left< T\left\{\frac{\delta F[\phi]}{\delta \phi^{\alpha}(x)} \right\}\right>.$$

Here the Euler-Lagrange expression is $\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}.$ However, one should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this Phys.SE question.

OP's equation is a special case of

$$\left< \Omega \left| T\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(y)}\right\}\right| \Omega \right>_J~=~0, \tag{A}$$ where the Euler-Lagrange expression is $\frac{\delta S[\phi;J]}{\delta \phi^{\alpha}(x)}.$

NB: Be aware that the time-ordering usually used in the literature is the covariant time-ordering $T_{\rm cov}$, i.e. time-differentiations inside its argument should be taken after/outside the usual time ordering $T$. This induces quantum corrections/contact terms, so that eq. (A) can be formally rewritten into the standard Schwinger-Dyson equations:

$$ \left< \Omega \left| T_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(y)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| T_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(y)} \right\}\right| \Omega \right>_J ~. \tag{B}$$ One should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this & this Phys.SE posts.

First of all, OP's eq. (3) needs a quantum correction given by the Schwinger-Dyson equation:

$$\tag{A} \left< T\left\{ F[\varphi]\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}\right\}\right>~=~i\hbar\left< T\left\{\frac{\delta F[\phi]}{\delta \phi^{\alpha}(x)} \right\}\right>.$$

Here the Euler-Lagrange expression is $\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}.$ However, one should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this Phys.SE question.

First of all, OP's eq. (3) needs a quantum correction given by the Schwinger-Dyson equation:

$$\tag{A} \left< T\left\{ F[\varphi]\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}\right\}\right>~=~i\hbar\left< T\left\{\frac{\delta F[\phi]}{\delta \phi^{\alpha}(x)} \right\}\right>.$$

Here the Euler-Lagrange expression is $\frac{\delta S[\phi]}{\delta \phi^{\alpha}(x)}.$ However, one should realize that this is just the beginning into a long discussion about operator ordering ambiguities, time ordering, and quantum correction, cf. e.g. this Phys.SE question.