Why does the ideal gas law apply when mass changes if it was derived at constant mass?

[Alwahsh]

A Question about the "General Gas Law" (PV/T)=nR

Homework Statement

The general gas law states that : PV/T = nR
where P is the pressure of the gas , V is the volume of the gas and T is the absolute temperature in Kelvin scale , n is the number of moles and R is the universal gas constant
My question is not really a problem but I can't understand something in the proof .

since PV/T = nR
therefore (P1V1/T1) = (P2V2/T2) We can say V1 = Density/Mass
therefore (P1M/Density1T1) = (P2M/Density2T2) At Constant Mass
therefore (P1/Density1T1) = (P2/Density2T2)

then this rule is taken to say :

Density = Mass/Volume
-At Constant Mass we say : (P1V1/T1) = (P2V2/T2) we exchanged density by Mass/volume and canceled the mass as it is constant .

-At Constant Volume we say : (P1/M1T1) = (P2/M2T2) we exchanged density by Mass/Volume and canceled the volume as it is constant .
My question is here , when the mass is changed , how can we use this rule although we said at the beginning of it "At Constant Mass" and where did the Mass that was in the very first rule go ??

I hope I've made myself clear .
Thanks

 

[Precursor]

Mass is still an entity in the ideal gas law. The variable n is the number of moles, and it equates to m/M, where m is the mass and M the molar mass.