Theorem imbibi

Description: The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025) (Proof shortened by Wolf Lammen, 5-Jan-2025)

	Ref	Expression
Assertion	imbibi	|- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.4	|- ( ( ph -> ( ph -> ps ) ) <-> ( ph -> ps ) )
2		imbi2	|- ( ( ( ph -> ps ) <-> ch ) -> ( ( ph -> ( ph -> ps ) ) <-> ( ph -> ch ) ) )
3	1 2	bitr3id	|- ( ( ( ph -> ps ) <-> ch ) -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
4	3	pm5.74rd	|- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) )