Metamath Proof Explorer

Theorem tbwlem2

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion tbwlem2 
|- ( ( ph -> ( ps -> F. ) ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )

Proof

Step Hyp Ref Expression
1 tbw-ax4 
 |-  ( F. -> ch )
2 tbw-ax1 
 |-  ( ( ps -> F. ) -> ( ( F. -> ch ) -> ( ps -> ch ) ) )
3 tbwlem1 
 |-  ( ( ( ps -> F. ) -> ( ( F. -> ch ) -> ( ps -> ch ) ) ) -> ( ( F. -> ch ) -> ( ( ps -> F. ) -> ( ps -> ch ) ) ) )
4 2 3 ax-mp 
 |-  ( ( F. -> ch ) -> ( ( ps -> F. ) -> ( ps -> ch ) ) )
5 1 4 ax-mp 
 |-  ( ( ps -> F. ) -> ( ps -> ch ) )
6 tbwlem1 
 |-  ( ( ( ps -> F. ) -> ( ps -> ch ) ) -> ( ps -> ( ( ps -> F. ) -> ch ) ) )
7 5 6 ax-mp 
 |-  ( ps -> ( ( ps -> F. ) -> ch ) )
8 tbw-ax1 
 |-  ( ( ph -> ( ps -> F. ) ) -> ( ( ( ps -> F. ) -> ch ) -> ( ph -> ch ) ) )
9 tbw-ax1 
 |-  ( ( ps -> ( ( ps -> F. ) -> ch ) ) -> ( ( ( ( ps -> F. ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
10 7 8 9 mpsyl 
 |-  ( ( ph -> ( ps -> F. ) ) -> ( ps -> ( ph -> ch ) ) )
11 tbw-ax1 
 |-  ( ( ps -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )
12 10 11 tbwsyl 
 |-  ( ( ph -> ( ps -> F. ) ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )