If your logical expressions aren't too strange, and there aren't too many, that's well suited to an SMT solver, such as Z3. You might, for example, create a ne

If your logical expressions aren't too strange, and there aren't too many, that's well suited to an SMT solver, such as Z3. You might, for example, create a new logical expression for each pair of your original expressions, joining each pair with AND, and then join those AND pairs with OR. Then ask the SMT solver to generate an unsatisfiability proof.