Theorem 19.45v

Description: Version of 19.45 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Assertion	19.45v	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists x\psi \right)$$

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	$$⊢\exists x\phi \leftrightarrow \phi$$
3	2	orbi1i	$$⊢\left(\exists x\phi \vee \exists x\psi \right)\leftrightarrow \left(\phi \vee \exists x\psi \right)$$
4	1 3	bitri	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists x\psi \right)$$