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Metamath Proof Explorer

Theorem ax2

Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion ax2 
|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Proof

Step Hyp Ref Expression
1 luklem7 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
2 luklem8 
 |-  ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) )
3 luklem6 
 |-  ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) )
4 luklem8 
 |-  ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5 3 4 ax-mp 
 |-  ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
6 2 5 luklem1 
 |-  ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
7 1 6 luklem1 
 |-  ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )