SAT $\in$ NPC

4.

Prove x $\in$ L' iff f(x) $\in$ L, where L' is CIRCUIT-SAT, L is SAT, and f is the construction above.

x $\in$ CIRCUIT-SAT $\rightarrow$ f(x) $\in$ SAT

If C has a satisfying assignment, then each wire is well-defined and the output is 1.

Therefore, each conjunct of $\phi$ is 1, and $\phi$ will evaluate to 1.

A satisfying assignment to $\phi$ yields a valid circuit C whose output is 1.