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Metamath Proof Explorer

Theorem luklem7

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 luklem7 
|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 luk-1 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
2 luklem5 
 |-  ( ps -> ( ( ps -> ch ) -> ps ) )
3 luk-1 
 |-  ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4 2 3 luklem1 
 |-  ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5 luklem6 
 |-  ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6 4 5 luklem1 
 |-  ( ps -> ( ( ps -> ch ) -> ch ) )
7 luk-1 
 |-  ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
8 6 7 ax-mp 
 |-  ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
9 1 8 luklem1 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )