Gittin 24 - June 9, 20 Sivan Daf Yomi for Women - Hadran

The Gemara brings support for Abaye's explanation that our Mishna is referring to a case where the get was brought from outside of Israel from a braita and from the language in our Mishna. How can Rav Yosef explain our Mishna according to his approach? In which case would a woman bringing her own get from outside of Israel need to say "in front of me it was written..."? Why would she need to do that if the moment she receives the get from her husband, she is divorced? The new perek starts delving into the issue of li'shma and gives 4 different cases where a get is not considered li'shma and would be invalid. What is the difference between all four cases?  What are possible ramifications from here to other promissory notes? Can these really be inferred from here? Even though a get that is not li'shma is not a valid get, does it disqualify a woman from marrying a kohen (in the event that her husband dies and she wishes to remarry)?