The gauge-invariant Yang-Mills and matter part is BRST invariant because a BRST variation is a gauge transformation with ghost-valued parameter. For the gauge-fixing fermion
(1)
the graded Leibniz rule gives
(2)
up to the common sign convention used to define the ghost term. This is precisely the gauge-fixing, auxiliary-field, and ghost sector of the displayed Lagrangian. Therefore
(3)

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