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Metamath Proof Explorer

Theorem impimprbi

Description: An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 , but <-> is a weaker operator than /\ . Note that an implication and its reverse can never be simultaneously false, because of pm2.521 . (Contributed by Wolf Lammen, 18-Dec-2023)

Ref Expression
Assertion impimprbi 
|- ( ( ph <-> ps ) <-> ( ( ph -> ps ) <-> ( ps -> ph ) ) )

Proof

Step Hyp Ref Expression
1 dfbi2 
 |-  ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2 pm5.1 
 |-  ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ( ph -> ps ) <-> ( ps -> ph ) ) )
3 1 2 sylbi 
 |-  ( ( ph <-> ps ) -> ( ( ph -> ps ) <-> ( ps -> ph ) ) )
4 impbi 
 |-  ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) )
5 pm2.521 
 |-  ( -. ( ph -> ps ) -> ( ps -> ph ) )
6 5 pm2.24d 
 |-  ( -. ( ph -> ps ) -> ( -. ( ps -> ph ) -> ( ph <-> ps ) ) )
7 4 6 bija 
 |-  ( ( ( ph -> ps ) <-> ( ps -> ph ) ) -> ( ph <-> ps ) )
8 3 7 impbii 
 |-  ( ( ph <-> ps ) <-> ( ( ph -> ps ) <-> ( ps -> ph ) ) )