Let $T$ or $T^*$ be an algebraically quasi-paranormal operator acting on Hilbert space. We prove: (i) Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$; (ii) $a$-Browder's theorem holds for $f(S)$ for every $S\prec T$ and $f\in H(\sigma(S))$; (iii) the spectral mapping theorem holds for the Weyl spectrum of $T$ and for the essential approximate point spectrum of $T$.