Is the following a valid alternative definition of overflow in twos complement arithmetic?

If the exclusive-OR of the carry bits into and out of the leftmost column is 1, then there is an overflow condition. Otherwise, there is not. 

 

12. The overflow rule was stated as follows: If two numbers are added, and they are both positive or both negative, then overflow occurs if and only if the result has the opposite sign. 
    
    There are four cases:
    •Both numbers positive (sign bit = 0) and no carry into the leftmost bit position:
    There is no carry out of the leftmost bit position, so the XOR is 0. The result has a
    sign bit = 0, so there is no overflow.
    
    •Both numbers positive and a carry into the leftmost bit position: There is no carry
    out of the leftmost position, so the XOR is 1. The result has a sign bit = 1, so there
    is overflow.
    
    •Both numbers negative and no carry into the leftmost position: There is a carry
    out of the leftmost position, so the XOR is 1. The result has a sign bit of 0, so there
    is overflow.
    
    •Both numbers negative and a carry into the leftmost position. There is a carry out
    of the leftmost position, so the XOR is 0. The result has a sign bit of 1, so there is
    no overflow.
    Therefore, the XOR result always agrees with the presence or absence of overflow.