Edit: Maybe it will make more intuitive sense if you put statements in.
"It is true that I'm not happy and that: I'm happy or rich (or both)." Do you see why this in all cases implies that "I'm rich"?
Edit 2: Sorry didn't notice the "without truth tables" part.
This might not be quite right, but here's my attempt
not p and (p or q) <-> (not p and p) or (not p and q) <-> not p and q
so we have
which is a tautology because the conclusion is the same as one of the premises.