Theorem merlem12 1578

Description: Step 28 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
merlem12	|- ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph )

Proof of Theorem merlem12

Step	Hyp	Ref	Expression
1		merlem5 1571	. . . 4 |- ( ( ch -> ch ) -> ( -. -. ch -> ch ) )
2		merlem2 1568	. . . 4 |- ( ( ( ch -> ch ) -> ( -. -. ch -> ch ) ) -> ( th -> ( -. -. ch -> ch ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( th -> ( -. -. ch -> ch ) )
4		merlem4 1570	. . 3 |- ( ( th -> ( -. -. ch -> ch ) ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) ) )
5	3, 4	ax-mp 5	. 2 |- ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) )
6		merlem11 1577	. 2 |- ( ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) ) -> ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) )
7	5, 6	ax-mp 5	1 |- ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> 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:  merlem13  1579