Theorem 19.42vvv

Description: Version of 19.42 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011) (Proof shortened by Wolf Lammen, 27-Aug-2023)

	Ref	Expression
Assertion	19.42vvv	$$⊢\exists x\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists x\exists y\exists z\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exdistr2	$$⊢\exists x\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \exists x\left(\phi \wedge \exists y\exists z\psi \right)$$
2		19.42v	$$⊢\exists x\left(\phi \wedge \exists y\exists z\psi \right)\leftrightarrow \left(\phi \wedge \exists x\exists y\exists z\psi \right)$$
3	1 2	bitri	$$⊢\exists x\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists x\exists y\exists z\psi \right)$$