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	$$⊢\phi \to \left(\chi \to \theta \right)$$
	anbiim.2	$$⊢\psi \to \left(\theta \to \chi \right)$$
Assertion	anbiim	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		anbiim.1	$$⊢\phi \to \left(\chi \to \theta \right)$$
2		anbiim.2	$$⊢\psi \to \left(\theta \to \chi \right)$$
3	1	adantr	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \to \theta \right)$$
4	2	adantl	$$⊢\left(\phi \wedge \psi \right)\to \left(\theta \to \chi \right)$$
5	3 4	impbid	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$