Theorem andiff

Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024)

	Ref	Expression
Hypotheses	andiff.1	$$⊢\phi \to \left(\chi \to \theta \right)$$
	andiff.2	$$⊢\psi \to \left(\theta \to \chi \right)$$
Assertion	andiff	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		andiff.1	$$⊢\phi \to \left(\chi \to \theta \right)$$
2		andiff.2	$$⊢\psi \to \left(\theta \to \chi \right)$$
3	1 2	anim12i	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(\chi \to \theta \right)\wedge \left(\theta \to \chi \right)\right)$$
4		dfbi2	$$⊢\left(\chi \leftrightarrow \theta \right)\leftrightarrow \left(\left(\chi \to \theta \right)\wedge \left(\theta \to \chi \right)\right)$$
5	3 4	sylibr	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$