Theorem alexbii

Description: Biconditional form of aleximi . (Contributed by BJ, 16-Nov-2020)

	Ref	Expression
Hypothesis	alexbii.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	alexbii	$$⊢\forall x\phi \to \left(\exists x\psi \leftrightarrow \exists x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		alexbii.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	biimpd	$$⊢\phi \to \left(\psi \to \chi \right)$$
3	2	aleximi	$$⊢\forall x\phi \to \left(\exists x\psi \to \exists x\chi \right)$$
4	1	biimprd	$$⊢\phi \to \left(\chi \to \psi \right)$$
5	4	aleximi	$$⊢\forall x\phi \to \left(\exists x\chi \to \exists x\psi \right)$$
6	3 5	impbid	$$⊢\forall x\phi \to \left(\exists x\psi \leftrightarrow \exists x\chi \right)$$