By Alexia Shurmur

LAS VEGAS, June 13 (Reuters) - Nevada Governor Brian Sandoval on Thursday vetoed Democratic-backed legislation that would have strengthened gun-control rules by requiring background checks on customers in all gun sales in the state, including private transactions.

The Republican governor said in a veto statement that the bill amounted to an erosion of Nevadans' constitutional right to bear arms that would do "little to prevent criminals from unlawfully obtaining firearms."

The bill, which narrowly passed in the state Senate in May and was approved by the Assembly in June, would have also required Nevada courts to send information about legal defendants who are found to be mentally ill to a national clearinghouse for all new gun purchases within five business days after the finding.

The veto came as Democratic lawmakers across the country pushed for stricter gun-control laws following mass shootings last year including the December massacre of 20 children and six adults at a Connecticut elementary school.

"Rather than sign sensible legislation that keeps guns out of the hands of convicted felons and the mentally ill, Governor Sandoval has decided to preserve the loopholes that they use to buy guns," New York Mayor Michael Bloomberg said in a statement from the Mayors Against Illegal Guns coalition.

Sandoval said in his veto message that he supported efforts to strengthen reporting requirements for the mentally ill but not the background check measure, which he said would have imposed "severe criminal penalties" on anyone who violates the required background check provision.

He said the bill, which the Nevada Sheriffs' and Chiefs Association criticized as unenforceable, would have barred anyone convicted of violating it from possessing a gun for two years, and indefinitely on a second offense. (Editing by Cynthia Johnston and Peter Cooney)

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