The activity that came to be named protein disulfide isomerase (PDI) was uncovered independently by Straub and Anfinsen in 1963. PDI was assigned an enzyme classification number (EC 184.108.40.206) in 1972:
EC 5… corresponds to “isomerase”
EC 5.3… corresponds to “intramolecular oxidoreductases”
EC 5.3.4. designation is for the category “transposing S-S bonds”.
So PDI is an oxidoreductase. In one catalytic mode, PDI undergoes cycles of oxidation and reduction – just swapping a single disulfide around:
To quote the stern Enzyme Commission: “oxidase will be used only for cases when O2 acts as an acceptor …”
Flavin dependent sulfhydryl oxidases are bona-fide oxidases. They function, after the transfer of reducing equivalents to the flavin prosthetic group, by transferring electrons to molecular oxygen:
So while PDI is an oxidoreductase, an isomerase and a foldase (and a wonderfully enigmatic enzyme all around) … it is never an oxidase!