How to simplify material implication?

Question

you know that the truth table for material implication is:

A | C | Y = A --> C
0 | 0 |       1
0 | 1 |       1
1 | 0 |       0
1 | 1 |       1

From this table we can deduce

A --> C = Y = ~A~C + ~AC + AC   (where ~X stands for NOT X)

But it is also well known that

A --> C = ~(A~C)

I can't reduce the 1st expression (~A~C + ~AC + AC) to the 2nd ( ~(A~C) ), can you show me through which steps can you obtain the 2nd from the 1st?

Thank you.

Answer 1

(~A~C + ~AC + AC)
(~A~C + ~AC) + AC
~A(~C + C) + AC
~A(T) + AC
~A + AC
~~(~A + AC)
~((~~A)~(AC))
~(A~(AC))
~(A(~A + ~C))
~(A~A + A~C)
~(F + A~C)
~(A~C)