Theorem 19.43OLD

Description: Obsolete proof of 19.43 . Do not delete as it is referenced on the mmrecent.html page and in conventions-labels . (Contributed by NM, 5-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ioran	$$⊢\neg \left(\phi \vee \psi \right)\leftrightarrow \left(\neg \phi \wedge \neg \psi \right)$$
2	1	albii	$$⊢\forall x\neg \left(\phi \vee \psi \right)\leftrightarrow \forall x\left(\neg \phi \wedge \neg \psi \right)$$
3		19.26	$$⊢\forall x\left(\neg \phi \wedge \neg \psi \right)\leftrightarrow \left(\forall x\neg \phi \wedge \forall x\neg \psi \right)$$
4		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
5		alnex	$$⊢\forall x\neg \psi \leftrightarrow \neg \exists x\psi$$
6	4 5	anbi12i	$$⊢\left(\forall x\neg \phi \wedge \forall x\neg \psi \right)\leftrightarrow \left(\neg \exists x\phi \wedge \neg \exists x\psi \right)$$
7	2 3 6	3bitri	$$⊢\forall x\neg \left(\phi \vee \psi \right)\leftrightarrow \left(\neg \exists x\phi \wedge \neg \exists x\psi \right)$$
8	7	notbii	$$⊢\neg \forall x\neg \left(\phi \vee \psi \right)\leftrightarrow \neg \left(\neg \exists x\phi \wedge \neg \exists x\psi \right)$$
9		df-ex	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \neg \forall x\neg \left(\phi \vee \psi \right)$$
10		oran	$$⊢\left(\exists x\phi \vee \exists x\psi \right)\leftrightarrow \neg \left(\neg \exists x\phi \wedge \neg \exists x\psi \right)$$
11	8 9 10	3bitr4i	$$⊢\exists x\left(\phi \vee \psi \right)\leftrightarrow \left(\exists x\phi \vee \exists x\psi \right)$$