Theorem merlem12 1418

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 1411	. . . 4 |- ((ch -> ch) -> (-. -. ch -> ch))
2		merlem2 1408	. . . 4 |- (((ch -> ch) -> (-. -. ch -> ch)) -> (th -> (-. -. ch -> ch)))
3	1, 2	ax-mp 5	. . 3 |- (th -> (-. -. ch -> ch))
4		merlem4 1410	. . 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 1417	. 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-meredith 1406

This theorem is referenced by:  merlem13  1419