Theorem mooran1

Description: "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	mooran1	$$⊢\left({\exists }^{*}x\phi \vee {\exists }^{*}x\psi \right)\to {\exists }^{*}x\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
2	1	moimi	$$⊢{\exists }^{*}x\phi \to {\exists }^{*}x\left(\phi \wedge \psi \right)$$
3		moan	$$⊢{\exists }^{*}x\psi \to {\exists }^{*}x\left(\phi \wedge \psi \right)$$
4	2 3	jaoi	$$⊢\left({\exists }^{*}x\phi \vee {\exists }^{*}x\psi \right)\to {\exists }^{*}x\left(\phi \wedge \psi \right)$$