A friend of mine just asked me how to prove that if $a=b$ then $a+c=b+c$, where $a,b$ and $c$ are real numbers, I'm not sure what I should answer. I have a book called introduction to logic and to the theory of the deductive sciences by Alfred Tarski, which is about propositional logic, and I remember reading that two things $a$ and $b$ are equal if any proposition that is true about $a$ is also true about $b$ and vice-versa. However I think this isn't very formal.
I haven't taken any set theory course, I think that another way to justify it is to say that sum is a function and since the ordered pairs $(a,c)$ and $(b,c)$ are equal then $+(a,c)=+(b,c)$. But I'm not too convinced.
If we use the standard axioms how would we justify $a+c=b+c$ using the mainstream axioms of today. I think there is something Zermelo-Frankl with choice. Would these be enough, what properties of the real numbers do we need? Can we prove it using the usual construction of the real numbers and Zermelo-Frankl?
As you can probably see I am not very knowledgeable about these topics, so I would like a delicate explanation.
Many thanks and regards.