Theorem moanimv

Description: Introduction of a conjunct into an at-most-one quantifier. Version of moanim requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995) Reduce axiom usage . (Revised by Wolf Lammen, 8-Feb-2023)

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

Proof

Step	Hyp	Ref	Expression
1		ibar	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
2	1	mobidv	$$⊢\phi \to \left({\exists }^{*}x\psi \leftrightarrow {\exists }^{*}x\left(\phi \wedge \psi \right)\right)$$
3		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
4	3	exlimiv	$$⊢\exists x\left(\phi \wedge \psi \right)\to \phi$$
5	2 4	moanimlem	$$⊢{\exists }^{*}x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \to {\exists }^{*}x\psi \right)$$