Theorem 19.41vvv

Description: Version of 19.41 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995)

	Ref	Expression
Assertion	19.41vvv	$$⊢\exists x\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \left(\exists x\exists y\exists z\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.41vv	$$⊢\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \left(\exists y\exists z\phi \wedge \psi \right)$$
2	1	exbii	$$⊢\exists x\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \exists x\left(\exists y\exists z\phi \wedge \psi \right)$$
3		19.41v	$$⊢\exists x\left(\exists y\exists z\phi \wedge \psi \right)\leftrightarrow \left(\exists x\exists y\exists z\phi \wedge \psi \right)$$
4	2 3	bitri	$$⊢\exists x\exists y\exists z\left(\phi \wedge \psi \right)\leftrightarrow \left(\exists x\exists y\exists z\phi \wedge \psi \right)$$