How do I prove that SAT in coNP implies NP=coNP?

Question

Is it true that if SAT is in coNP then its also coNP-complete (because it is NP-complete)?

Answer 1

If SAT is in coNP then since coSAT is coNP-complete, SAT can be reduced to coSAT in polytime. The same reduction also reduces coSAT to SAT. Since coSAT is coNP-complete, every problem in coNP can be reduced in polytime to coSAT, and so to SAT. This shows that SAT is coNP-complete.