Name of the arithmetic property if a=b then a+c = b+c? Properties of arithmetic operations such multiplication and division have names. For example:
$a + b = b + a$ (commutativity)
$(a + b) + c = a + (b + c)$ associativity 
and so on
is there a name for
if $a=b$ then $a+c = b+c$ ?
 A: I don't think it has a name. It's just a special case of $a=b\implies f(a)=f(b)$.
A: Adding $c$ to both sides of an equation is a so-called "equivalence transformation" ; that is a transformation that does not change the set of solutions of the equation.
A: I don't think it has a name. The converse rule, however:
$$
a + c = b + c\implies a = b
$$
is called cancellation. You could argue that your rule is in some sense cancellation for subtraction, but it takes more work than it's worth, in my opinion.
A: Well, the generalization would be monotonicity of addition:
$$a\leq b\Rightarrow a+c\leq b+c.$$
Note that if $a\leq b\wedge b\leq a\Rightarrow a=b$ and 
$a\leq b :\Leftrightarrow \exists c: a+c=b.$
Here if you take as underlying set the ring of integers, the field of rational, real or complex numbers, you are fine.
A: The book Introduction to Mathematical Logic by Elliot Mendelson, which was the textbook for my first course in mathematical logic, has a section on "First-Order Theories with Equality" starting on page 75. The book has two requirements for a theory with equality. The second requirement is that this is a theorem:
$$x=y \supset (\mathscr{A}(x,x) \supset \mathscr{A}(x,x))$$
where $x$ and $y$ are any variables, $\mathscr{A}(x,x)$ is any well-formed formula, and $\mathscr{A}(x,y)$ arises from $\mathscr{A}(x,x)$ by replacing some, but not necessarily all, free occurences of $x$ by $y$, with the proviso that $y$ is free for the occurences of $x$ which it replaces. (And of course, the "$\supset$" symbol means implication.) The name of this axiom schema is

Substitutivity of Equality

Your property is a particular theorem included in this schema. The schema is a generalization of @J.G.'s answer, but it has a name.
A: See : 
Transforming equations : rules governing the use of '<=>' and '=>' . and Mauro Allegranza's answer. 
What follows is simply a development of J.G 's answer. 
Define a function f , say from the set of real numbers  to the set of real numbers ,  such that f(x) = x+2;  that is, in terms of relation, f is the relation such that  : 
f = {(x,y) | y = x + 2 }. 
A relation R is a function iff : 
no two different ordered pairs belonging to R have the same first element. 
In other words, a relation R  is a function iff : 
all orders pairs belonging to R that have the same first element also have the same second element. 
Now, the ordered pairs ( a, z) and (b, z') such that z = a+2 and z'= b+2 belong to the function f . 
These two pairs have the same first element, since, by hypothesis: a = b. 
Consequently, they also have the same second element, which means that z = z'. 
According to the definitions of z and of z', it means that : 
                             a+2 = b+2 

