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Metamath Proof Explorer

Theorem luklem2

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 luklem2 
|- ( ( ph -> -. ps ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )

Proof

Step Hyp Ref Expression
1 luk-1 
 |-  ( ( ph -> -. ps ) -> ( ( -. ps -> ch ) -> ( ph -> ch ) ) )
2 luk-3 
 |-  ( ps -> ( -. ps -> ch ) )
3 luk-1 
 |-  ( ( ps -> ( -. ps -> ch ) ) -> ( ( ( -. ps -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
4 2 3 ax-mp 
 |-  ( ( ( -. ps -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
5 1 4 luklem1 
 |-  ( ( ph -> -. ps ) -> ( ps -> ( ph -> ch ) ) )
6 luk-1 
 |-  ( ( ps -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )
7 5 6 luklem1 
 |-  ( ( ph -> -. ps ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )