Zevachim 51A

מטהרת טריפתה מטומאתה; אף מליקה, שמכשרתה באכילה ⁠– תטהר טריפתה מטומאתה.

and it purifies its tereifa from its impurity, so too its pinching, which permits bird offerings with regard to consumption, should purify its tereifa from its impurity.

רבי יוסי אומר: דיה כנבלת בהמה טהורה ⁠– ששחיטתה מטהרתה, ולא מליקתה.

R' Yosei says: Although one can derive from the case of an animal that slaughter purifies the tereifa of a bird from its impurity, that derivation cannot be extended to pinching. The same restriction that applies to every a fortiori inference, namely, that a halakha derived by means of an a fortiori inference is no more stringent than the source from which it is derived, applies here: It is sufficient for the halakhic status of the carcass of a bird that is a tereifa to be like that of the carcass of an animal of a kosher species that is a tereifa; i.e., that only its slaughter purifies it, but not its pinching.

ולא היא; התם תיהוי היא, משחיטה דחולין קאתיין.

The Talmud rejects this proof: And that is not so. Let it remain there, i.e., one cannot learn from it, as that is a case that comes from the slaughter of non-sacred animals. The halakha of the pinching of a consecrated bird is derived through a paradigm from the halakha of the slaughter of a non-sacred bird, and the halakha of the slaughter of a non-sacred bird is derived through an a fortiori inference from the halakha of the slaughter of a non-sacred animal. Outside of the realm of consecrated matters there is no question that a matter derived via one of the hermeneutical principles can then teach its halakha via another principle. The entire question under discussion is only with regard to the realm of consecrated matters.