ΔE 21.0, loose: different enough that you should pick on purpose. Dioxazine Violet is noticeably darker, more saturated, cooler than Phoenician Purple.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 21). Dioxazine Violet is noticeably darker, more saturated, cooler than Phoenician Purple.
The same difference applies the other way round: Phoenician Purple is noticeably lighter, more muted, warmer.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Phoenician Purple on the colour wheel See Dioxazine Violet on the colour wheel
All matches: Phoenician Purple equivalents · Dioxazine Violet equivalents