How do I simplify this boolean expression? I have constructed truth tables to prove that:
$ABC + ABC'+ AB'C +A'BC = AB+AC+BC$

How do I prove it by simplifying the expression? I know that I can simplify:  $ABC + ABC' = AB(C+C')=AB$. However I can't repeat this for $AB'C$ or $A'BC$, to get the answer I desire. 

I'm fairly new to boolean algebra and have tried to use the basic identities to figure it out, but can't seem to get there.
$
\begin{align}
ABC + ABC' + AB'C + A'BC = \\
ABC + ABC + ABC + ABC' + AB'C + A'BC =  &amp; &amp; &amp; \text{(2x Idempotency)}\\
AB(C + C') + ABC + ABC + AB'C + A'BC = &amp; &amp; &amp;\text{(Distribution)}\\
AB + ABC + ABC + AB'C + A'BC = &amp; &amp; &amp; \text{(Negation)}\\
AB + (AB + AB')C + (AB + A'B)C = &amp; &amp; &amp; \text{(2x Distribution)} \\
AB + (A(B+B'))C + ((A+A')B)C = &amp; &amp; &amp; \text {(2x Distribution)} \\
AB + AC + BC &amp; &amp; &amp; \text{(2x Negation)}
\end{align}
$

I'm not proficient at this kind of derivations so please correct any notation/terminology