Theorem tbwlem1 1630

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbwlem1	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

Proof of Theorem tbwlem1

Step	Hyp	Ref	Expression
1		tbw-ax2 1626	. . . 4 |- ( ps -> ( ( ps -> ch ) -> ps ) )
2		tbw-ax1 1625	. . . 4 |- ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
3	1, 2	tbwsyl 1629	. . 3 |- ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4		tbw-ax1 1625	. . . 4 |- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5		tbw-ax3 1627	. . . 4 |- ( ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6	4, 5	tbwsyl 1629	. . 3 |- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
7	3, 6	tbwsyl 1629	. 2 |- ( ps -> ( ( ps -> ch ) -> ch ) )
8		tbw-ax1 1625	. 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
9		tbw-ax1 1625	. 2 |- ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
10	7, 8, 9	mpsyl 68	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:  tbwlem2  1631  tbwlem4  1633  tbwlem5  1634  re1luk3  1637