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Metamath Proof Explorer

Theorem tbw-bijust

Description: Justification for tbw-negdf . (Contributed by Anthony Hart, 15-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion tbw-bijust 
|- ( ( ph <-> ps ) <-> ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) )

Proof

Step Hyp Ref Expression
1 dfbi1 
 |-  ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
2 pm2.21 
 |-  ( -. ( ps -> ph ) -> ( ( ps -> ph ) -> F. ) )
3 2 imim2i 
 |-  ( ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
4 id 
 |-  ( -. ( ps -> ph ) -> -. ( ps -> ph ) )
5 falim 
 |-  ( F. -> -. ( ps -> ph ) )
6 4 5 ja 
 |-  ( ( ( ps -> ph ) -> F. ) -> -. ( ps -> ph ) )
7 6 imim2i 
 |-  ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
8 3 7 impbii 
 |-  ( ( ( ph -> ps ) -> -. ( ps -> ph ) ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
9 8 notbii 
 |-  ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) <-> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
10 pm2.21 
 |-  ( -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) )
11 ax-1 
 |-  ( -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) ) )
12 falim 
 |-  ( F. -> ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) ) )
13 11 12 ja 
 |-  ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) ) )
14 13 pm2.43i 
 |-  ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
15 10 14 impbii 
 |-  ( -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) <-> ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) )
16 1 9 15 3bitri 
 |-  ( ( ph <-> ps ) <-> ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) )