SAT $\in$ NPC

Proof:

1.

SAT $\in$ NP
Replace each variable with 0 or 1 as specified by the certificate and evaluate (poly-time).

2.

Select L' = CIRCUIT-SAT

3.

Reduction from CIRCUIT-SAT to SAT.
Straight-forward technique of computing the formula of each gate output as a combination of the input formulae may cause exponential instantiations of a variable as outputs are copied to multiple inputs. Instead, let each gate output be a variable.
AND the output variable with expressions for each gate describing the equivalence between the gate's output and input variables.

Example

Circuit C

Formula

$$\phi \;=\; x_6 \;\wedge\; (x_4 \;\leftrightarrow\; (x_1 \wedg... ...g x_3) \;\wedge\; (x_6 \;\leftrightarrow\; (x_4 \;\vee\; x_5))$$

Constructing this formula takes polynomial time.