Replace each variable with 0 or 1 as specified by the certificate and evaluate (poly-time).
Select L' = CIRCUIT-SAT
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.
Constructing this formula takes polynomial time.