Is this a valid method of proof?

Question

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by cancellation, conclude that $b = c$, so this proves that $y = a + c$?

Answer 1

Yes it proves, provided $b=c$ i.e.

$y=a+c $ and $y=a+b$

For the both equations to be true we assume $a+b=a+c$

which leads us to $b=c$

So your equation is true under the condition $a=c$

Answer 2

Not really accurate I guess. Your method is legitimate under real field assumption which is the usual real number system if you care too much in details. Since in real number system $a-a=0$ and $ a+0=a$. However in arbitrary system one could start with $a+b=a+c$ but cancellation rule may not be applied I guess.