Theorem merlem11

Description: Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merlem11	|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		meredith	|- ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) )
2		merlem10	|- ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
3		merlem10	|- ( ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) -> ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) )
4	2 3	ax-mp	|- ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
5	1 4	ax-mp	|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )