Metamath Proof Explorer

Theorem merlem4

Description: Step 8 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	merlem4	\|- ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		meredith	\|- ( ( ( ( ( ph -> ph ) -> ( -. th -> -. th ) ) -> th ) -> ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) )
2		merlem3	\|- ( ( ( ( ( ( ph -> ph ) -> ( -. th -> -. th ) ) -> th ) -> ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) ) -> ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) ) )
3	1 2	ax-mp	\|- ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) )

Step

Hyp

Ref

Expression

meredith

 |-  ( ( ( ( ( ph -> ph ) -> ( -. th -> -. th ) ) -> th ) -> ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) )

merlem3

 |-  ( ( ( ( ( ( ph -> ph ) -> ( -. th -> -. th ) ) -> th ) -> ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) ) -> ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) ) )

1 2

ax-mp

 |-  ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) )