Consider a certain formal system with only axiom Excluded Middle -$EM$

[![enter image description here][1]][1]

and 18 inference rules:

9 implicative ruules (clearly not independent)

[![enter image description here][2]][2]

and 9 tautological rules:

[![enter image description here][3]][3]

If we have substitution at hands as well but we are restricted no to use conditional proof.

Is this particular system complete?

I always believed that the system is complete. But once I decided to prove or disprove completeness I stuck. I neither can find any reference neither nor can proof completeness.

If someone is familiar with this system I would be grateful to have some reference or proof. [1]: https://i.sstatic.net/o5WuK.png [2]: https://i.sstatic.net/6nGnu.png [3]: https://i.sstatic.net/fUZCG.png

Consider a certain formal system with only axiom Excluded Middle -$EM$

and 18 inference rules:

9 implicative ruules (clearly not independent)

and 9 tautological rules:

If we have substitution at hands as well but we are restricted no to use conditional proof.

Is this particular system complete?

I always believed that the system is complete. But once I decided to prove or disprove completeness I stuck. I neither can find any reference neither nor can proof completeness.

If someone is familiar with this system I would be grateful to have some reference or proof.