No. Maybe if you write it on paper, with the usual logical symbol... it's an "if then" implication not an "if and only if": a equals b => a's hash code is equal to b's hash code. (TRUE) This can be written also: a's hash code not equal to b's hash code => a is not equal to b (TRUE).
But you can't deduce the opposite implication: a's H.C. == b's H.C. => a equals b (FALSE) This (false) statement can be written also as: a not equal b => a's H.C. != b's H.C. Maybe written this way it's easier to see because it's false (mind, I mean the implication is false; it may be true in some cases, but you can't be sure).