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Metamath Proof Explorer

Theorem nanbi

Description: Biconditional in terms of alternative denial. (Contributed by Jeff Hoffman, 19-Nov-2007) (Proof shortened by Wolf Lammen, 27-Jun-2020)

Ref Expression
Assertion nanbi 
|- ( ( ph <-> ps ) <-> ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) ) )

Proof

Step Hyp Ref Expression
1 dfbi3 
 |-  ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) )
2 df-or 
 |-  ( ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <-> ( -. ( ph /\ ps ) -> ( -. ph /\ -. ps ) ) )
3 df-nan 
 |-  ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) )
4 3 bicomi 
 |-  ( -. ( ph /\ ps ) <-> ( ph -/\ ps ) )
5 nannot 
 |-  ( -. ph <-> ( ph -/\ ph ) )
6 nannot 
 |-  ( -. ps <-> ( ps -/\ ps ) )
7 5 6 anbi12i 
 |-  ( ( -. ph /\ -. ps ) <-> ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) )
8 4 7 imbi12i 
 |-  ( ( -. ( ph /\ ps ) -> ( -. ph /\ -. ps ) ) <-> ( ( ph -/\ ps ) -> ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) ) )
9 1 2 8 3bitri 
 |-  ( ( ph <-> ps ) <-> ( ( ph -/\ ps ) -> ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) ) )
10 nannan 
 |-  ( ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) ) <-> ( ( ph -/\ ps ) -> ( ( ph -/\ ph ) /\ ( ps -/\ ps ) ) ) )
11 9 10 bitr4i 
 |-  ( ( ph <-> ps ) <-> ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) ) )