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 ) ) )