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Metamath Proof Explorer

Theorem luklem6

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion luklem6 
|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

Proof

Step Hyp Ref Expression
1 luk-1 
 |-  ( ( ph -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) )
2 luklem5 
 |-  ( -. ( ph -> ps ) -> ( -. ps -> -. ( ph -> ps ) ) )
3 luklem2 
 |-  ( ( -. ps -> -. ( ph -> ps ) ) -> ( ( ( -. ps -> ps ) -> ps ) -> ( ( ph -> ps ) -> ps ) ) )
4 luklem4 
 |-  ( ( ( ( -. ps -> ps ) -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) )
5 3 4 luklem1 
 |-  ( ( -. ps -> -. ( ph -> ps ) ) -> ( ( ph -> ps ) -> ps ) )
6 2 5 luklem1 
 |-  ( -. ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) )
7 luk-1 
 |-  ( ( -. ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( -. ( ph -> ps ) -> ( ph -> ps ) ) ) )
8 6 7 ax-mp 
 |-  ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( -. ( ph -> ps ) -> ( ph -> ps ) ) )
9 luk-1 
 |-  ( ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( -. ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ( -. ( ph -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) )
10 8 9 ax-mp 
 |-  ( ( ( -. ( ph -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
11 luklem4 
 |-  ( ( ( ( -. ( ph -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
12 10 11 ax-mp 
 |-  ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) )
13 1 12 luklem1 
 |-  ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )