Theorem ax2 1592

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

Assertion

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

Proof of Theorem ax2

Step	Hyp	Ref	Expression
1		luklem7 1589	. 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
2		luklem8 1590	. . 3 |- ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) )
3		luklem6 1588	. . . 4 |- ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) )
4		luklem8 1590	. . . 4 |- ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5	3, 4	ax-mp 5	. . 3 |- ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
6	2, 5	luklem1 1583	. 2 |- ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
7	1, 6	luklem1 1583	1 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)