If $A \vdash a$ and $B \vdash b$ then $A \cup B \vdash a$ and $A \cup B \vdash b$? Question about the empty set and axioms. Is it true/accepted that: 
If $A \vdash a$ and $B \vdash b$ then $A \cup B \vdash a$ and $A \cup B \vdash b$?
Is there a name for this concept? Does it require its own meta-proof or is it just a concept we accept and use intuitively?
Because we normally say an axiom is of the form $\vdash \varphi$ or equivalently $\emptyset \vdash \varphi$, and in logical proofs we are allowed to use axioms even when the lefthand side differs. For instance for some arbitrary set $\Delta$ and axiom $\varphi$ we're allowed to say $\Delta \vdash \varphi$ even though the axiom is not "coming from" or "proved from" $\Delta$ but rather the empty set, and so in some ways this is like saying $\Delta \cup \emptyset \vdash \varphi$.
Is this why we're allowed to write things like $\Delta \vdash \varphi$? Because since $\Delta = \Delta \cup \emptyset$ and $\emptyset \vdash \varphi$ we can write $\Delta \vdash \varphi$?
 A: Given a (possibly empty) set of formulas $\Delta$, a derivation of $\Delta \vdash A$ in a Hilbert system is also a derivation of $\Delta \cup \Gamma \vdash A$ in such Hilbert system, for any set of formulas $\Gamma$. This property (a trivial meta-theorem for Hilbert systems) is often called weakening.
Its proof is immediate. Indeed, a derivation $\Delta \vdash A$ is a finite sequence of formulas in which the last formula is $A$ and each formula is either a formula in $\Delta$, or is a logical axiom or is obtained from previous formulas in the sequence by an inference rule. Since $\Delta \subseteq \Delta \cup \Gamma$ for any set of formulas $\Gamma$, the same derivation can be seen also as a derivation of $\Delta \cup \Gamma \vdash A$ in the Hilbert system: simply, the formulas in $\Gamma$ are not used in the derivation, but there is nothing to prevent to add them to the set of assumptions. 
The property of weakening intuitively means that if you can derive something from some hypotheses, then you can derive the same thing adding more hypotheses to the previous ones. The name of this property is weakening because claiming that $\Delta \cup \Gamma \vdash A$ is derivable is weaker (or less informative) than claiming that $\Delta \vdash A$: the more hypotheses you need to derive something, the weaker is your result. 
