Theorem tbwlem4 1473

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem tbwlem4

Step	Hyp	Ref	Expression
1		tbw-ax4 1468	. . . . 5 |- ( F. -> F. )
2		tbw-ax1 1465	. . . . . 6 |- ((ps -> F. ) -> (( F. -> F. ) -> (ps -> F. )))
3		tbwlem1 1470	. . . . . 6 |- (((ps -> F. ) -> (( F. -> F. ) -> (ps -> F. ))) -> (( F. -> F. ) -> ((ps -> F. ) -> (ps -> F. ))))
4	2, 3	ax-mp 5	. . . . 5 |- (( F. -> F. ) -> ((ps -> F. ) -> (ps -> F. )))
5	1, 4	ax-mp 5	. . . 4 |- ((ps -> F. ) -> (ps -> F. ))
6		tbwlem1 1470	. . . 4 |- (((ps -> F. ) -> (ps -> F. )) -> (ps -> ((ps -> F. ) -> F. )))
7	5, 6	ax-mp 5	. . 3 |- (ps -> ((ps -> F. ) -> F. ))
8		tbw-ax1 1465	. . . 4 |- (((ph -> F. ) -> ps) -> ((ps -> ((ps -> F. ) -> F. )) -> ((ph -> F. ) -> ((ps -> F. ) -> F. ))))
9		tbwlem1 1470	. . . 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 5	. . 3 |- ((ps -> ((ps -> F. ) -> F. )) -> (((ph -> F. ) -> ps) -> ((ph -> F. ) -> ((ps -> F. ) -> F. ))))
11	7, 10	ax-mp 5	. 2 |- (((ph -> F. ) -> ps) -> ((ph -> F. ) -> ((ps -> F. ) -> F. )))
12		tbwlem2 1471	. . 3 |- (((ph -> F. ) -> ((ps -> F. ) -> F. )) -> ((((ph -> F. ) -> ph) -> ph) -> ((ps -> F. ) -> ph)))
13		tbwlem3 1472	. . 3 |- (((((ph -> F. ) -> ph) -> ph) -> ((ps -> F. ) -> ph)) -> ((ps -> F. ) -> ph))
14	12, 13	tbwsyl 1469	. 2 |- (((ph -> F. ) -> ((ps -> F. ) -> F. )) -> ((ps -> F. ) -> ph))
15	11, 14	tbwsyl 1469	1 |- (((ph -> F. ) -> ps) -> ((ps -> F. ) -> ph))

Syntax hints:   -> wi 4   F. wfal 1317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320

This theorem is referenced by:  tbwlem5  1474  re1luk2  1476