Theorem luklem7 1589

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

Assertion

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

Proof of Theorem luklem7

Step	Hyp	Ref	Expression
1		luk-1 1580	. 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
2		luklem5 1587	. . . . 5 |- ( ps -> ( ( ps -> ch ) -> ps ) )
3		luk-1 1580	. . . . 5 |- ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4	2, 3	luklem1 1583	. . . 4 |- ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5		luklem6 1588	. . . 4 |- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6	4, 5	luklem1 1583	. . 3 |- ( ps -> ( ( ps -> ch ) -> ch ) )
7		luk-1 1580	. . 3 |- ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
8	6, 7	ax-mp 5	. 2 |- ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
9	1, 8	luklem1 1583	1 |- ( ( ph -> ( ps -> ch ) ) -> ( 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:  luklem8  1590  ax2  1592