Theorem merlem7 1413

Description: Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merlem7	|- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))

Proof of Theorem merlem7

Step	Hyp	Ref	Expression
1		merlem4 1410	. 2 |- ((ps -> ch) -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))
2		merlem6 1412	. . . 4 |- ((((ch -> ta) -> (-. th -> -. ps)) -> th) -> (((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)))
3		ax-meredith 1406	. . . 4 |- (((((ch -> ta) -> (-. th -> -. ps)) -> th) -> (((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph))) -> (((((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)) -> ch) -> (ps -> ch)))
4	2, 3	ax-mp 5	. . 3 |- (((((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)) -> ch) -> (ps -> ch))
5		ax-meredith 1406	. . 3 |- ((((((((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)) -> -. ph) -> (-. ch -> -. ph)) -> ch) -> (ps -> ch)) -> (((ps -> ch) -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))) -> (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))))
6	4, 5	ax-mp 5	. 2 |- (((ps -> ch) -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))) -> (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th))))
7	1, 6	ax-mp 5	1 |- (ph -> (((ps -> ch) -> th) -> (((ch -> ta) -> (-. th -> -. ps)) -> th)))

Syntax hints:  -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-meredith 1406

This theorem is referenced by:  merlem8  1414