I believe it was first proved by Kleiman (you only need to assume fixed dimension and degree), see

S. Kleiman, Exp XIII in A. Grothendieck et al., Theorie des Intersections et Theoreme de Riemann-Roch (SGA 6), Lecture Notes in Math No. 225, Springer-Verlag, Heidelberg (1971).

I believe it was first proved by Kleiman (you only need to assume fixed dimension and degree, no need to assume a subvariety of some fixed $\mathbb P^n$), see Corollary 6.11

S. Kleiman, Exp XIII in A. Grothendieck et al., Theorie des Intersections et Theoreme de Riemann-Roch (SGA 6), Lecture Notes in Math No. 225, Springer-Verlag, Heidelberg (1971).

I believe it was first proved by Kleiman, see

S. Kleiman, Exp XIII in A. Grothendieck et al., Theorie des Intersections et Theoreme de Riemann-Roch (SGA 6), Lecture Notes in Math No. 225, Springer-Verlag, Heidelberg (1971).

I believe it was first proved by Kleiman (you only need to assume fixed dimension and degree), see

S. Kleiman, Exp XIII in A. Grothendieck et al., Theorie des Intersections et Theoreme de Riemann-Roch (SGA 6), Lecture Notes in Math No. 225, Springer-Verlag, Heidelberg (1971).