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Metamath Proof Explorer

Theorem tbw-negdf

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.)

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

Proof

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