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	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
	anbiim.2	⊢ ( 𝜓 → ( 𝜃 → 𝜒 ) )
Assertion	anbiim	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		anbiim.1	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
2		anbiim.2	⊢ ( 𝜓 → ( 𝜃 → 𝜒 ) )
3	1	adantr	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 → 𝜃 ) )
4	2	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜃 → 𝜒 ) )
5	3 4	impbid	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) )