Theorem tbw-negdf 1464

Description: The definition of negation, in terms of -> and F.. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tbw-negdf	|- (((-. ph -> (ph -> F. )) -> (((ph -> F. ) -> -. ph) -> F. )) -> F. )

Proof of Theorem tbw-negdf

Step	Hyp	Ref	Expression
1		pm2.21 100	. . 3 |- (-. ph -> (ph -> F. ))
2		ax-1 6	. . . . 5 |- (-. ph -> ((ph -> F. ) -> -. ph))
3		falim 1328	. . . . 5 |- ( F. -> ((ph -> F. ) -> -. ph))
4	2, 3	ja 153	. . . 4 |- ((ph -> F. ) -> ((ph -> F. ) -> -. ph))
5	4	pm2.43i 43	. . 3 |- ((ph -> F. ) -> -. ph)
6	1, 5	impbii 180	. 2 |- (-. ph <-> (ph -> F. ))
7		tbw-bijust 1463	. 2 |- ((-. ph <-> (ph -> F. )) <-> (((-. ph -> (ph -> F. )) -> (((ph -> F. ) -> -. ph) -> F. )) -> F. ))
8	6, 7	mpbi 199	1 |- (((-. ph -> (ph -> F. )) -> (((ph -> F. ) -> -. ph) -> F. )) -> F. )

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   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:  re1luk2  1476  re1luk3  1477