Theorem luklem4 664

Description: Lemma for rederiving standard propositional axioms from Lukasiewicz'.

Assertion

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

Proof of Theorem luklem4

Step	Hyp	Ref	Expression
1		luk-2 659	. . . 4 |- ((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph))
2		luk-2 659	. . . . 5 |- ((-. ph -> ph) -> ph)
3		luklem3 663	. . . . 5 |- (((-. ph -> ph) -> ph) -> (((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph)) -> (-. ps -> ((-. ph -> ph) -> ph))))
4	2, 3	ax-mp 6	. . . 4 |- (((-. ((-. ph -> ph) -> ph) -> ((-. ph -> ph) -> ph)) -> ((-. ph -> ph) -> ph)) -> (-. ps -> ((-. ph -> ph) -> ph)))
5	1, 4	ax-mp 6	. . 3 |- (-. ps -> ((-. ph -> ph) -> ph))
6		luk-1 658	. . 3 |- ((-. ps -> ((-. ph -> ph) -> ph)) -> ((((-. ph -> ph) -> ph) -> ps) -> (-. ps -> ps)))
7	5, 6	ax-mp 6	. 2 |- ((((-. ph -> ph) -> ph) -> ps) -> (-. ps -> ps))
8		luk-2 659	. 2 |- ((-. ps -> ps) -> ps)
9	7, 8	luklem1 661	1 |- ((((-. ph -> ph) -> ph) -> ps) -> ps)

Syntax hints:  -. wn 1   -> wi 2

This theorem is referenced by:  luklem5 665  luklem6 666  ax3 671

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

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