Theorem merlem1 1567

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.)

Assertion

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

Proof of Theorem merlem1

Step	Hyp	Ref	Expression
1		meredith 1566	. . 3 |- ( ( ( ( ( -. ph -> ps ) -> ( -. ( -. ta -> -. ch ) -> -. -. ( -. ph -> ps ) ) ) -> ( -. ta -> -. ch ) ) -> ta ) -> ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) )
2		meredith 1566	. . 3 |- ( ( ( ( ( ( -. 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 5	. 2 |- ( ( ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) -> ( -. ph -> ps ) ) -> ( ch -> ( -. ph -> ps ) ) )
4		meredith 1566	. 2 |- ( ( ( ( ( ta -> -. ph ) -> ( -. ( -. ph -> ps ) -> -. ph ) ) -> ( -. ph -> ps ) ) -> ( ch -> ( -. ph -> ps ) ) ) -> ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) ) )
5	3, 4	ax-mp 5	1 |- ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) )

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  merlem2  1568  merlem5  1571  luk-3  1582