> "No Correlation does imply non-causation"

You're wrong. Random variables can be uncorrelated but not independent [1].

In general, you can't apply "algebra" to logic as you attempted above and explained below. Logic is very precise. To "prove" that `A => B` is equivalent `~B => ~A`, you use truth tables:

    | A | B | A => B | ~B => ~A |
    +---+---+--------+----------+
    | 0 | 0 |    1   |     1    |
    | 0 | 1 |    1   |     1    |
    | 1 | 0 |    0   |     0    |
    | 1 | 1 |    1   |     1    |

Your statements above aren't equivalent; the first is a tautology (it's always true), whereas the second is standard implication, which can be false.

[1] http://en.wikipedia.org/wiki/Normally_distributed_and_uncorr...