Theorem merlem11 1577

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.)

Assertion

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

Proof of Theorem merlem11

Step	Hyp	Ref	Expression
1		meredith 1566	. 2 |- ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) )
2		merlem10 1576	. . 3 |- ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
3		merlem10 1576	. . 3 |- ( ( ( 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 5	. 2 |- ( ( ( ( ( ( ph -> ph ) -> ( -. ph -> -. ph ) ) -> ph ) -> ph ) -> ( ( ph -> ph ) -> ( ph -> ph ) ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
5	1, 4	ax-mp 5	1 |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

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:  merlem12  1578  merlem13  1579  luk-2  1581  luk-3  1582