Theorem merlem3 1569

Description: Step 7 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
merlem3	|- ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) )

Proof of Theorem merlem3

Step	Hyp	Ref	Expression
1		merlem2 1568	. . . 4 |- ( ( ( -. ch -> -. ch ) -> ( -. ch -> -. ch ) ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) )
2		merlem2 1568	. . . 4 |- ( ( ( ( -. ch -> -. ch ) -> ( -. ch -> -. ch ) ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) ) -> ( ( ( ( ch -> ph ) -> ( -. ps -> -. ps ) ) -> ps ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( ( ( ( ch -> ph ) -> ( -. ps -> -. ps ) ) -> ps ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) )
4		meredith 1566	. . 3 |- ( ( ( ( ( ch -> ph ) -> ( -. ps -> -. ps ) ) -> ps ) -> ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) ) -> ( ( ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) -> ch ) -> ( ps -> ch ) ) )
5	3, 4	ax-mp 5	. 2 |- ( ( ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) -> ch ) -> ( ps -> ch ) )
6		meredith 1566	. 2 |- ( ( ( ( ( ph -> ph ) -> ( -. ch -> -. ch ) ) -> ch ) -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) ) )
7	5, 6	ax-mp 5	1 |- ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) )

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:  merlem4  1570  merlem6  1572