I'm trying to prove the following problem:
Prove that if $P=NP$ then there is a polynomial time algorithm for the following problem:
INPUT: A boolean formula $\phi$.
OUTPUT: A satisfying assignment of $\phi$ if $\phi$ is satisfiable. If $\phi$ is unsatisfiable, return $false$.
If $P=NP$ then $SAT$ can be decided in polynomial time. Run $SAT$ on $\phi$. If it returns $false$, reject. Else, nondeterministically select a boolean assignment. If it satisfies $\phi$, return it.
The algorithm is in $NP$, but because $P=NP$, it is also in $P$.
Is my proof correct? I'm asking because the textbook gave a different answer, and I want to know if my proof is correct, and if not, where does it fall?