Theorem bj-3exbi

Description: Closed form of 3exbii . (Contributed by BJ, 6-May-2019)

	Ref	Expression
Assertion	bj-3exbi	$$⊢\forall x\forall y\forall z\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\exists y\exists z\phi \leftrightarrow \exists x\exists y\exists z\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exbi	$$⊢\forall z\left(\phi \leftrightarrow \psi \right)\to \left(\exists z\phi \leftrightarrow \exists z\psi \right)$$
2	1	2alimi	$$⊢\forall x\forall y\forall z\left(\phi \leftrightarrow \psi \right)\to \forall x\forall y\left(\exists z\phi \leftrightarrow \exists z\psi \right)$$
3		bj-2exbi	$$⊢\forall x\forall y\left(\exists z\phi \leftrightarrow \exists z\psi \right)\to \left(\exists x\exists y\exists z\phi \leftrightarrow \exists x\exists y\exists z\psi \right)$$
4	2 3	syl	$$⊢\forall x\forall y\forall z\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\exists y\exists z\phi \leftrightarrow \exists x\exists y\exists z\psi \right)$$