Theorem 19.44

Description: Theorem 19.44 of Margaris p. 90. See 19.44v for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypothesis	19.44.1	$$⊢Ⅎx\psi$$
Assertion	19.44	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.44.1	$$⊢Ⅎx\psi$$
2		19.43	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists x\psi \right)$$
3	1	19.9	$$⊢\exists x\psi \leftrightarrow \psi$$
4	3	orbi2i	$$⊢\left(\exists x\phi \vee \exists x\psi \right)\leftrightarrow \left(\exists x\phi \vee \psi \right)$$
5	2 4	bitri	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \psi \right)$$