Suppose that P != NP. Then there exists 3SAT formulas such that their satisfiability is computationally "evil" (i.e, the satisfiability can be exponentially hard to determine in the size of the formula).
Primality testing can be reduced to the satisfiability of a multiplication circuit, which can be reduced to the satisfiability of a 3SAT formula. But "Primes is in P". The satisfiability of this formula cannot be "evil", since it can be reduced to AKS primality test for example.
Integer factorization can be achieved by finding a satisfying assignment to the same formula. This can be achieved by choosing a variable, and then testing the satisfiability again. Choosing a variable, simplifies the formula through unit propagation. The new formula is a subset of the previous one. Now suddenly "we don't know" whether this formula can be solved in polynomial time.
Isn't this a contradiction? Is it possible that the subset of a formula is harder to solve than the formula itself?