Theorem anbiim

Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024) (Proof shortened by Wolf Lammen, 7-May-2025)

	Ref	Expression
Hypotheses	anbiim.1	|- ( ph -> ( ch -> th ) )
	anbiim.2	|- ( ps -> ( th -> ch ) )
Assertion	anbiim	|- ( ( ph /\ ps ) -> ( ch <-> th ) )

Proof

Step	Hyp	Ref	Expression
1		anbiim.1	|- ( ph -> ( ch -> th ) )
2		anbiim.2	|- ( ps -> ( th -> ch ) )
3	1	adantr	|- ( ( ph /\ ps ) -> ( ch -> th ) )
4	2	adantl	|- ( ( ph /\ ps ) -> ( th -> ch ) )
5	3 4	impbid	|- ( ( ph /\ ps ) -> ( ch <-> th ) )