Theorem re1luk2 1636

Description: luk-2 1581 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1luk2	|- ( ( -. ph -> ph ) -> ph )

Proof of Theorem re1luk2

Step	Hyp	Ref	Expression
1		tbw-negdf 1624	. . . 4 |- ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. )
2		tbw-ax2 1626	. . . . 5 |- ( ( ( ( ph -> F. ) -> -. ph ) -> F. ) -> ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) )
3		tbwlem4 1633	. . . . 5 |- ( ( ( ( ( ph -> F. ) -> -. ph ) -> F. ) -> ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) ) -> ( ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) -> ( ( ph -> F. ) -> -. ph ) ) )
4	2, 3	ax-mp 5	. . . 4 |- ( ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) -> ( ( ph -> F. ) -> -. ph ) )
5	1, 4	ax-mp 5	. . 3 |- ( ( ph -> F. ) -> -. ph )
6		tbw-ax1 1625	. . 3 |- ( ( ( ph -> F. ) -> -. ph ) -> ( ( -. ph -> ph ) -> ( ( ph -> F. ) -> ph ) ) )
7	5, 6	ax-mp 5	. 2 |- ( ( -. ph -> ph ) -> ( ( ph -> F. ) -> ph ) )
8		tbw-ax3 1627	. 2 |- ( ( ( ph -> F. ) -> ph ) -> ph )
9	7, 8	tbwsyl 1629	1 |- ( ( -. ph -> ph ) -> ph )

Syntax hints:   -. wn 3   -> 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: (None)