Theorem 19.43

Description: Theorem 19.43 of Margaris p. 90. (Contributed by NM, 12-Mar-1993) (Proof shortened by Wolf Lammen, 27-Jun-2014)

	Ref	Expression
Assertion	19.43	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-or	$$⊢\left(\phi \vee \psi \right)\leftrightarrow \left(\neg \phi \to \psi \right)$$
2	1	exbii	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \exists x\left(\neg \phi \to \psi \right)$$
3		19.35	$$⊢\exists x\left(\neg \phi \to \psi \right)\leftrightarrow \left(\forall x\neg \phi \to \exists x\psi \right)$$
4		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
5	4	imbi1i	$$⊢\left(\forall x\neg \phi \to \exists x\psi \right)\leftrightarrow \left(\neg \exists x\phi \to \exists x\psi \right)$$
6	2 3 5	3bitri	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\neg \exists x\phi \to \exists x\psi \right)$$
7		df-or	$$⊢\left(\exists x\phi \vee \exists x\psi \right)\leftrightarrow \left(\neg \exists x\phi \to \exists x\psi \right)$$
8	6 7	bitr4i	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists x\psi \right)$$