Theorem merlem1

Description: Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merlem1	|- ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) )

Proof

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