Theorem tbwlem1

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.)

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

Proof

Step	Hyp	Ref	Expression
1		tbw-ax2	|- ( ps -> ( ( ps -> ch ) -> ps ) )
2		tbw-ax1	|- ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
3	1 2	tbwsyl	|- ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4		tbw-ax1	|- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5		tbw-ax3	|- ( ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6	4 5	tbwsyl	|- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
7	3 6	tbwsyl	|- ( ps -> ( ( ps -> ch ) -> ch ) )
8		tbw-ax1	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
9		tbw-ax1	|- ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
10	7 8 9	mpsyl	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )