Theorem 19.42v

Description: Version of 19.42 with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

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

Proof

Step	Hyp	Ref	Expression
1		19.41v	$$⊢\exists x\left(\psi \wedge \phi \right)\leftrightarrow \left(\exists x\psi \wedge \phi \right)$$
2		exancom	$$⊢\exists x\left(\phi \wedge \psi \right)\leftrightarrow \exists x\left(\psi \wedge \phi \right)$$
3		ancom	$$⊢\left(\phi \wedge \exists x\psi \right)\leftrightarrow \left(\exists x\psi \wedge \phi \right)$$
4	1 2 3	3bitr4i	$$⊢\exists x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \exists x\psi \right)$$