fight (somebody/something) tooth and nail
to use a lot of effort to oppose someone or achieve something

Idiom Definitions for 'Fight tooth and nail'

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If someone will fight tooth and nail for something, they will not stop at anything to get what they want. ('Fight tooth and claw' is an alternative.)

To fight violently is to fight "tooth and nail," with every "tooth" and every "nail." Example: "That woman fought tooth and nail during the war to protect her children." Animals sometimes fight violently with every tooth (in their mouth) and every nail (on their claws). Example: "That cat is terrified (very afraid) of cars; try to get him in a car and he'll fight you tooth and nail." To fight "tooth and nail" is to fight completely and with everything you have. Note: "Tooth" and "nail" are both singular in "tooth and nail." Example: "That was a tough match; they fought us tooth and nail!"

böyle yorumluyorum...

Verdiğiniz ifadenin "dişimizle tırnağımızla uğraşıp didinmek", "dişimizle tırnağımızla yapmak" anlamında kullanıldığını varsayarsak, buna yakın anlam taşıyan bir ifade biliyorum:

"to go at it with hammer and tongs" - çok büyük çaba sarfederek bir şeyi yapmaya çalışmak:

http://www.thefreedictionary.com/hammer and tongs
http://en.wikipedia.org/wiki/Hammer_&_Tongs

Ama belirtmem gerek ki, bu ifadenin bir diğer anlamı da, şiddetli bir şekilde tartışmak:

http://www.usingenglish.com/reference/idioms/hammer and tong...

Bu ifade bazı kaynaklarda "to go at it hammer and tongs" olarak kullanılıyor:

http://www.urbandictionary.com/define.php?term=hammer and to...

i.e. by using everything possible against the opposite part.

I'd use one of these as the case might be.