Theorem tbw-negdf 1624

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		pm2.21 120	. . 3 |- ( -. ph -> ( ph -> F. ) )
2		ax-1 6	. . . . 5 |- ( -. ph -> ( ( ph -> F. ) -> -. ph ) )
3		falim 1498	. . . . 5 |- ( F. -> ( ( ph -> F. ) -> -. ph ) )
4	2, 3	ja 173	. . . 4 |- ( ( ph -> F. ) -> ( ( ph -> F. ) -> -. ph ) )
5	4	pm2.43i 52	. . 3 |- ( ( ph -> F. ) -> -. ph )
6	1, 5	impbii 199	. 2 |- ( -. ph <-> ( ph -> F. ) )
7		tbw-bijust 1623	. 2 |- ( ( -. ph <-> ( ph -> F. ) ) <-> ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) )
8	6, 7	mpbi 220	1 |- ( ( ( -. ph -> ( ph -> F. ) ) -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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:  re1luk2  1636  re1luk3  1637