Theorem 19.45

Description: Theorem 19.45 of Margaris p. 90. See 19.45v for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypothesis	19.45.1	$$⊢Ⅎx\phi$$
Assertion	19.45	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.45.1	$$⊢Ⅎx\phi$$
2		19.43	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists x\psi \right)$$
3	1	19.9	$$⊢\exists x\phi \leftrightarrow \phi$$
4	3	orbi1i	$$⊢\left(\exists x\phi \vee \exists x\psi \right)\leftrightarrow \left(\phi \vee \exists x\psi \right)$$
5	2 4	bitri	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\phi \vee \exists x\psi \right)$$