the axiom system given in the slides that you point to is correct (sound and complete) for LTL. You can also find several variations on LTL proof systems, with different combinations of operators, in our book
Fred Kröger, Stephan Merz. Temporal Logic and State Systems. Texts in Theoretical Computer Science, Springer, 2008.
What you wrote is an axiom for the weak until or unless operator, typically written W, not to be confused with the (strong) until operator denoted by U.
Here is a system of axioms for LTL (slide 7)
Can you validate its accuracy?
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