Theorem 19.44v

Description: Version of 19.44 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Assertion	19.44v	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.43	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists x\psi \right)$$
2		19.9v	$$⊢\exists x\psi \leftrightarrow \psi$$
3	2	orbi2i	$$⊢\left(\exists x\phi \vee \exists x\psi \right)\leftrightarrow \left(\exists x\phi \vee \psi \right)$$
4	1 3	bitri	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \psi \right)$$