Every bill presented to the mayor and returned to the council with the approval of the mayor shall become an ordinance, and every bill presented as aforesaid, but returned with his objections thereto, shall stand reconsidered. The council shall cause the objections of the mayor to be entered at large upon the journal, and proceed at its convenience to consider the question pending, which shall be in this form: Shall the bill pass, the objections of the mayor thereto notwithstanding? The votes on this question shall be taken by yeas and nays and the names entered upon the journal, and if two-thirds of all the members-elect shall vote in the affirmative, the president shall certify the fact on the roll, and the bill thus certified shall be deposited with the proper officer, and shall become an ordinance in the same manner and with like effect as if it had received the approval of the mayor. The mayor shall have power to sign or veto any ordinance passed by the city council, and shall also possess the power to approve all or any portion of the general appropriation bill, or to veto any item or all of the same; provided, that should he neglect or refuse to sign any ordinance and return the same with his objections, in writing, at the next regular meeting of the council, the same shall become a law without his signature.