Combinational Logic - Need help with proof by rewrite In my Computer Logic class we were assigned the following problem: 


  Complete a truth table that has $3$ inputs $(A, B,C)$ and one output
  $(F)$.   $F$ is        asserted whenever $B$ or $C$ are asserted, but
  deasserted if both $B$ and $C$ are asserted.  $F$ is asserted whenever $A$ is
  asserted and $C$ is deasserted.  $F$ is deasserted in all other cases.


After which we are supposed to 


  Determine the sum-of-products equation for $F$.  Show the terms instead
  of the minterm abbreviations (show $ABC$ instead of m7).


Finally, we are supposed to:


  Prove that $F$ from question $2$ is equivalent to $F = BC' + B'C + AC'$
  using proof by rewrite.  HINT: remember the uniting law is the key to
  minimization.


Here is what I have:

1)

A    B    C    F
0    0    0    0
0    0    1    1
0    1    0    1
0    1    1    0
1    0    0    1
1    0    1    1
1    1    0    1
1    1    1    1


2) $F = A'B'C + A'BC' + AB'C' + AB'C + ABC'$

Number three is what is throwing me. I am using the axioms of boolean algebra, such as the Uniting Law, but so far all I have been able to do is the following:

3) 
    A’B’C + A’BC’ + AB’C’ + AB’C + ABC’ = BC’ + B’C + AC’

A’B’C + A’BC’ + AB’C’ + AB’C + ABC’

A’B’C + AB’C + A’BC’ + AB’C’  + ABC’            Commutative Law

B’C + A’BC’ + AB’C’ + ABC’                      Uniting Law

A’BC’ + ABC’ + B’C + AB’C’                      Commutative Law

BC’ + B’C + AB’C’                               Uniting Law


That's as far as I am able to get. Did I screw up the truth table, resulting in incorrect minterms, or did I miss something? 

Appreciate the help!
The last line is wrong : 
1    1    1    1
Should be : 
1    1    1    0

But the equations following are correct.

For the simplification :
BC’ + B’C + AB’C’ = BC' + B'.(C+AC') = BC'+B'.(C+A)=BC'+B'C+B'A

(I don't know the name of the rules used, there is some associativity, "X+X'Y = X+Y", etc...)

--- edit ---

Let's prove that X+X'Y = X+Y

Using :


A = A.B + A.B'
A = A+A
A + AB = A.(1+B) = A
A + A' = 1


X+X'Y = XY+XY'+X'Y = Y(X+X') + XY' = Y + XY'

Repeating (A=A+A) :

X+X'Y = X+X'Y + X+X'Y = X+X'Y + Y+XY' 

= X + Y + X'Y + XY' 

= X+XY' + Y+X'Y

= X + Y