True-Sat problem is NP-complete

TRUE-SAT = {boolean expressions E in conjunctive normal form that
                   (1) are true when all variables are set true, AND ALSO
                   (2) have some other truth assignment that makes E true}
            This can also be stated (in Garey-Johnson notation) as:  
             Instance: Set of clauses C₁,C₂,....C_k over the boolean variables u₁,u₂,....u_m  
             Question:
                     (1) are all C_i true when u₁,u₂,....u_m are set to true, and
                     (2) is there another satisfying truth assignment (not all variables set to true)?

First, Show TRUE_SAT is in NP. (This is left as an exercise)
Then show TRUE-SAT is NP-hard.
If an expression is not true when all variables are true, then it is surely not in TRUE-SAT.

To show TRUE-SAT is NP-complete, we reduce SAT to it. Suppose we are given an expression E with variables a,b,c.

Convert E to E' as follows:
Test if E is true when a = T, b=T and c=T.
If so, we know E is satisfiable, then add (a U a') to E.
                 E' = E  U ( a U a' )
                 // E satisfies first condition. a U a' satisfies the second condition.
Otherwise
                E' = E U (  a U b U c )
                //Adding  a U b U c satisfies first condition. If E is in SAT, that satisfies the second condition.
 Therefore, TRUE_SAT accepts only when E is in SAT. Hence TRUE-SAT is reducible to SAT and TRUE-SAT is NP-complete.