Theorem tbwlem4 15122

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.

Assertion

Ref	Expression
tbwlem4	|- (((ph -> F. ) -> ps) -> ((ps -> F. ) -> ph))

Proof of Theorem tbwlem4

Step	Hyp	Ref	Expression
1		tbw-ax4 15117	. . . . 5 |- ( F. -> F. )
2		tbw-ax1 15114	. . . . . 6 |- ((ps -> F. ) -> (( F. -> F. ) -> (ps -> F. )))
3		tbwlem1 15119	. . . . . 6 |- (((ps -> F. ) -> (( F. -> F. ) -> (ps -> F. ))) -> (( F. -> F. ) -> ((ps -> F. ) -> (ps -> F. ))))
4	2, 3	ax-mp 7	. . . . 5 |- (( F. -> F. ) -> ((ps -> F. ) -> (ps -> F. )))
5	1, 4	ax-mp 7	. . . 4 |- ((ps -> F. ) -> (ps -> F. ))
6		tbwlem1 15119	. . . 4 |- (((ps -> F. ) -> (ps -> F. )) -> (ps -> ((ps -> F. ) -> F. )))
7	5, 6	ax-mp 7	. . 3 |- (ps -> ((ps -> F. ) -> F. ))
8		tbw-ax1 15114	. . . 4 |- (((ph -> F. ) -> ps) -> ((ps -> ((ps -> F. ) -> F. )) -> ((ph -> F. ) -> ((ps -> F. ) -> F. ))))
9		tbwlem1 15119	. . . 4 |- ((((ph -> F. ) -> ps) -> ((ps -> ((ps -> F. ) -> F. )) -> ((ph -> F. ) -> ((ps -> F. ) -> F. )))) -> ((ps -> ((ps -> F. ) -> F. )) -> (((ph -> F. ) -> ps) -> ((ph -> F. ) -> ((ps -> F. ) -> F. )))))
10	8, 9	ax-mp 7	. . 3 |- ((ps -> ((ps -> F. ) -> F. )) -> (((ph -> F. ) -> ps) -> ((ph -> F. ) -> ((ps -> F. ) -> F. ))))
11	7, 10	ax-mp 7	. 2 |- (((ph -> F. ) -> ps) -> ((ph -> F. ) -> ((ps -> F. ) -> F. )))
12		tbwlem2 15120	. . 3 |- (((ph -> F. ) -> ((ps -> F. ) -> F. )) -> ((((ph -> F. ) -> ph) -> ph) -> ((ps -> F. ) -> ph)))
13		tbwlem3 15121	. . 3 |- (((((ph -> F. ) -> ph) -> ph) -> ((ps -> F. ) -> ph)) -> ((ps -> F. ) -> ph))
14	12, 13	tbwsyl 15118	. 2 |- (((ph -> F. ) -> ((ps -> F. ) -> F. )) -> ((ps -> F. ) -> ph))
15	11, 14	tbwsyl 15118	1 |- (((ph -> F. ) -> ps) -> ((ps -> F. ) -> ph))

Syntax hints:   -> wi 3   F. wfal 1565

This theorem is referenced by:  tbwlem5 15123  re1luk2 15125

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

This theorem depends on definitions:  df-bi 232  df-tru 1566  df-fal 1567

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