Is there a law that you can add or multiply to both sides of an equation? It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.
Why isn't this an axiom of arithmetic, like the commutative law or associative law?
Or is it a consequence of some other axiom of arithmetic?
Thanks!
Edit: I understand the intuitive meaning of equality. Answers that stated that $a = b$ means they are the same number or object make sense but what I'm asking is if there is an explicit law of replacement that allows us to make this intuitive truth a valid mathematical deduction. For example is there an axiom of Peano's Axioms or some other axiomatic system that allows for adding or multiplying both sides of an equation by the same number? 
In all the texts I've come across I've never seen an axiom that states if $a = b$ then $a + c = b + c$. I have however seen if $a < b$ then $a + c < b + c$. In my view $<$ and $=$ are similar so the absence of a definition for equality is strange.

This is an axiom of predicate logic. For example, check out this list of the axioms in predicate calulus, intended to be an ambient logic for ZFC set theory. Note axioms 13 and 14:
$$\vdash x=y\to (x\in z\to y\in z)$$
$$\vdash x=y\to (z\in x\to z\in y)$$
In set theory, the only basic atomic formulas are of the form $x=y$ or $x\in y$, so this, together with transitivity of equality (axiom 8), which will allow you to prove
$$\vdash x=y\to (x=z\to y=z)$$
$$\vdash x=y\to (z=x\to z=y),$$
is sufficient to prove by induction that for any predicate of the language $\varphi(x)$, it is a theorem that
$$\vdash x=y\to(\varphi(x)\leftrightarrow \varphi(y)).$$
And once you define class terms via $x\in\{x\mid\varphi\}\leftrightarrow\varphi$, you can prove
$$\vdash x=y\to A(x)=A(y)$$
for any class term $A(x)$ by converting it to the statement $$\vdash x=y\to (z\in A(x)\leftrightarrow z\in A(y))$$
using extensionality and applying the theorem above for predicates. This demonstrates how the rule $x=y\to f(x)=f(y)$ gets translated into a rigorous proof in a formal system.
