Theorem luklem4 1426

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
luklem4	|- ((((-. ph -> ph) -> ph) -> ps) -> ps)

Proof of Theorem luklem4

Step	Hyp	Ref	Expression
1		luk-2 1421	. . . 4 |- ((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph))
2		luk-2 1421	. . . . 5 |- ((-. ph -> ph) -> ph)
3		luklem3 1425	. . . . 5 |- (((-. ph -> ph) -> ph) -> (((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph)) -> (-. ps -> ((-. ph -> ph) -> ph))))
4	2, 3	ax-mp 5	. . . 4 |- (((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph)) -> (-. ps -> ((-. ph -> ph) -> ph)))
5	1, 4	ax-mp 5	. . 3 |- (-. ps -> ((-. ph -> ph) -> ph))
6		luk-1 1420	. . 3 |- ((-. ps -> ((-. ph -> ph) -> ph)) -> ((((-. ph -> ph) -> ph) -> ps) -> (-. ps -> ps)))
7	5, 6	ax-mp 5	. 2 |- ((((-. ph -> ph) -> ph) -> ps) -> (-. ps -> ps))
8		luk-2 1421	. 2 |- ((-. ps -> ps) -> ps)
9	7, 8	luklem1 1423	1 |- ((((-. ph -> ph) -> ph) -> ps) -> ps)

Syntax hints:  -. wn 3   -> wi 4

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

This theorem is referenced by:  luklem5  1427  luklem6  1428  ax3  1433