Theorem tbwlem4 1633

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 1628	. . . . 5 |- ( F. -> F. )
2		tbw-ax1 1625	. . . . . 6 |- ( ( ps -> F. ) -> ( ( F. -> F. ) -> ( ps -> F. ) ) )
3		tbwlem1 1630	. . . . . 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 1630	. . . 4 |- ( ( ( ps -> F. ) -> ( ps -> F. ) ) -> ( ps -> ( ( ps -> F. ) -> F. ) ) )
7	5, 6	ax-mp 5	. . 3 |- ( ps -> ( ( ps -> F. ) -> F. ) )
8		tbw-ax1 1625	. . . 4 |- ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> ( ( ps -> F. ) -> F. ) ) -> ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) ) )
9		tbwlem1 1630	. . . 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 1631	. . 3 |- ( ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) -> ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ( ( ps -> F. ) -> ph ) ) )
13		tbwlem3 1632	. . 3 |- ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ( ( ps -> F. ) -> ph ) ) -> ( ( ps -> F. ) -> ph ) )
14	12, 13	tbwsyl 1629	. 2 |- ( ( ( ph -> F. ) -> ( ( ps -> F. ) -> F. ) ) -> ( ( ps -> F. ) -> ph ) )
15	11, 14	tbwsyl 1629	1 |- ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> F. ) -> ph ) )

Syntax hints:   -> wi 4   F. wfal 1488

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 197  df-tru 1486  df-fal 1489

This theorem is referenced by:  tbwlem5  1634  re1luk2  1636