Theorem tbwlem5 1634

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
tbwlem5	|- ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph )

Proof of Theorem tbwlem5

Step	Hyp	Ref	Expression
1		tbw-ax2 1626	. . . 4 |- ( ph -> ( ps -> ph ) )
2		tbw-ax1 1625	. . . 4 |- ( ( ps -> ph ) -> ( ( ph -> F. ) -> ( ps -> F. ) ) )
3	1, 2	tbwsyl 1629	. . 3 |- ( ph -> ( ( ph -> F. ) -> ( ps -> F. ) ) )
4		tbwlem1 1630	. . 3 |- ( ( ph -> ( ( ph -> F. ) -> ( ps -> F. ) ) ) -> ( ( ph -> F. ) -> ( ph -> ( ps -> F. ) ) ) )
5	3, 4	ax-mp 5	. 2 |- ( ( ph -> F. ) -> ( ph -> ( ps -> F. ) ) )
6		tbwlem4 1633	. 2 |- ( ( ( ph -> F. ) -> ( ph -> ( ps -> F. ) ) ) -> ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph ) )
7	5, 6	ax-mp 5	1 |- ( ( ( ph -> ( ps -> F. ) ) -> 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:  re1luk3  1637