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Blood Feud lets two creatures fight each other. Is it similar to normal combat?
As I understand it, after a combat you use up your power and toughness. If there are 3/4 and 2/2, and they combat each other, the first one uses 2 powers to kill the other and has remaining 1 power. Its toughness is used up to be 2.
Does the same thing happen and the effect remains till the end of the turn after Blood Feud?
This is not a correct understanding of either combat or fighting.
"Fight" is a special keyword that simply means "the 2 chosen creatures deal damage to the other equal to their power." Neither power nor toughness is ever "used up." If a 2/4 creature fights another 2/4 creature, both will still be a 2/4 creature after that. The only difference will be that they will both be a 2/4 creature with 2 damage on them.
In combat, power and toughness are also not "used up". The attacking and blocking creature will deal damage to the other equal to its power.