Theorem luklem4 941

Description: Used to rederive standard propositional axioms from Lukasiewicz'.

Assertion

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

Proof of Theorem luklem4

Step	Hyp	Ref	Expression
1		luk-2 936	. . . 4 |- ((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph))
2		luk-2 936	. . . . 5 |- ((-. ph -> ph) -> ph)
3		luklem3 940	. . . . 5 |- (((-. ph -> ph) -> ph) -> (((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph)) -> (-. ps -> ((-. ph -> ph) -> ph))))
4	2, 3	ax-mp 7	. . . 4 |- (((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph)) -> (-. ps -> ((-. ph -> ph) -> ph)))
5	1, 4	ax-mp 7	. . 3 |- (-. ps -> ((-. ph -> ph) -> ph))
6		luk-1 935	. . 3 |- ((-. ps -> ((-. ph -> ph) -> ph)) -> ((((-. ph -> ph) -> ph) -> ps) -> (-. ps -> ps)))
7	5, 6	ax-mp 7	. 2 |- ((((-. ph -> ph) -> ph) -> ps) -> (-. ps -> ps))
8		luk-2 936	. 2 |- ((-. ps -> ps) -> ps)
9	7, 8	luklem1 938	1 |- ((((-. ph -> ph) -> ph) -> ps) -> ps)

Syntax hints:  -. wn 2   -> wi 3

This theorem is referenced by:  luklem5 942  luklem6 943  ax3 948

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

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