As it seems that on the research level notation is the biggest problem for some of people, I will share a link to classical notation of Łukasiewicz L3 system, where deductive theorem i classical meaning does not holshohold ( but modified version - holds) - just take a look here: L3 Łukasiewicz logic and here: Łukasiewicz L3 system and reference:
Bergmann, Merrie (2008). An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems. Cambridge University Press. p. 114. ISBN 978-0-521-88128-9.