Metamath Proof Explorer

Theorem tbwlem4

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 tbwlem4 
|- ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> F. ) -> ph ) )

Proof

Step Hyp Ref Expression
1 tbw-ax4 
 |-  ( F. -> F. )
2 tbw-ax1 
 |-  ( ( ps -> F. ) -> ( ( F. -> F. ) -> ( ps -> F. ) ) )
3 tbwlem1 
 |-  ( ( ( ps -> F. ) -> ( ( F. -> F. ) -> ( ps -> F. ) ) ) -> ( ( F. -> F. ) -> ( ( ps -> F. ) -> ( ps -> F. ) ) ) )
4 2 3 ax-mp 
 |-  ( ( F. -> F. ) -> ( ( ps -> F. ) -> ( ps -> F. ) ) )
5 1 4 ax-mp 
 |-  ( ( ps -> F. ) -> ( ps -> F. ) )
6 tbwlem1 
 |-  ( ( ( ps -> F. ) -> ( ps -> F. ) ) -> ( ps -> ( ( ps -> F. ) -> F. ) ) )
7 5 6 ax-mp 
 |-  ( ps -> ( ( ps -> F. ) -> F. ) )
8 tbw-ax1 
 |-  ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> ( ( ps -> F. ) -> F. ) ) -> ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) ) )
9 tbwlem1 
 |-  ( ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> ( ( ps -> F. ) -> F. ) ) -> ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) ) ) -> ( ( ps -> ( ( ps -> F. ) -> F. ) ) -> ( ( ( ph -> F. ) -> ps ) -> ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ps -> ( ( ps -> F. ) -> F. ) ) -> ( ( ( ph -> F. ) -> ps ) -> ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) ) )
11 7 10 ax-mp 
 |-  ( ( ( ph -> F. ) -> ps ) -> ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) )
12 tbwlem2 
 |-  ( ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) -> ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ( ( ps -> F. ) -> ph ) ) )
13 tbwlem3 
 |-  ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ( ( ps -> F. ) -> ph ) ) -> ( ( ps -> F. ) -> ph ) )
14 12 13 tbwsyl 
 |-  ( ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) -> ( ( ps -> F. ) -> ph ) )
15 11 14 tbwsyl 
 |-  ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> F. ) -> ph ) )