  The aim of this short paper is to establish a spectral algebra analog of the
Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the
context of algebraic structures that can be described as algebras over an
operad $\mathcal{O}$ in symmetric spectra. Our main result is that completion
with respect to topological Quillen homology (or TQ-completion, for short)
preserves homotopy fibration sequences provided that the base and total
$\mathcal{O}$-algebras are connected. Our argument essentially boils down to
proving that the natural map from the homotopy fiber to its TQ-completion tower
is a pro-$\pi_*$ isomorphism. More generally, we also show that similar results
remain true if we replace "homotopy fibration sequence" with "homotopy pullback
square."
