Metamath Proof Explorer

Theorem re1luk2

Description: luk-2 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion re1luk2 
|- ( ( -. ph -> ph ) -> ph )

Proof

Step Hyp Ref Expression
1 tbw-negdf 
 |-  ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. )
2 tbw-ax2 
 |-  ( ( ( ( ph -> F. ) -> -. ph ) -> F. ) -> ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) )
3 tbwlem4 
 |-  ( ( ( ( ( ph -> F. ) -> -. ph ) -> F. ) -> ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) ) -> ( ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) -> ( ( ph -> F. ) -> -. ph ) ) )
4 2 3 ax-mp 
 |-  ( ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) -> ( ( ph -> F. ) -> -. ph ) )
5 1 4 ax-mp 
 |-  ( ( ph -> F. ) -> -. ph )
6 tbw-ax1 
 |-  ( ( ( ph -> F. ) -> -. ph ) -> ( ( -. ph -> ph ) -> ( ( ph -> F. ) -> ph ) ) )
7 5 6 ax-mp 
 |-  ( ( -. ph -> ph ) -> ( ( ph -> F. ) -> ph ) )
8 tbw-ax3 
 |-  ( ( ( ph -> F. ) -> ph ) -> ph )
9 7 8 tbwsyl 
 |-  ( ( -. ph -> ph ) -> ph )