Theorem merlem7

Description: Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merlem7	|- ( ph -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) )

Proof

Step	Hyp	Ref	Expression
1		merlem4	|- ( ( ps -> ch ) -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) )
2		merlem6	|- ( ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) -> ( ( ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) -> -. ph ) -> ( -. ch -> -. ph ) ) )
3		meredith	|- ( ( ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) -> ( ( ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) -> -. ph ) -> ( -. ch -> -. ph ) ) ) -> ( ( ( ( ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) -> -. ph ) -> ( -. ch -> -. ph ) ) -> ch ) -> ( ps -> ch ) ) )
4	2 3	ax-mp	|- ( ( ( ( ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) -> -. ph ) -> ( -. ch -> -. ph ) ) -> ch ) -> ( ps -> ch ) )
5		meredith	|- ( ( ( ( ( ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) -> -. ph ) -> ( -. ch -> -. ph ) ) -> ch ) -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) ) -> ( ph -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) ) ) )
6	4 5	ax-mp	|- ( ( ( ps -> ch ) -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) ) -> ( ph -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) ) )
7	1 6	ax-mp	|- ( ph -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) -> ( -. th -> -. ps ) ) -> th ) ) )