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For coordinate $i$, keep the other coordinates fixed and propose $y_i$ from the full conditional $\pi (\cdot \mid x_{-i})$. This is a Metropolis–Hastings algorithm proposal on that coordinate fibre, with deterministic equality $y_{-i}=x_{-i}$. Its acceptance ratio isThus the coordinate update has acceptance probability one and is exactly a single-coordinate Gibbs sampler update.

A systematic sweep through coordinates $1,\dots ,p$ is the composition of these kernels. Its joint transition density isThe already updated coordinates are new values, and the not-yet-updated coordinates are old values. Each coordinate kernel preserves $\pi$, so their composition also preserves it. Systematic Gibbs sampling need not be reversible: each individual coordinate kernel is reversible, but their ordered composition need not be reversible; invariance is the property needed here. A random-scan mixture of the coordinate kernels is reversible.