Theorem moanim

Description: Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv . (Contributed by NM, 3-Dec-2001) (Proof shortened by Wolf Lammen, 24-Dec-2018)

	Ref	Expression
Hypothesis	moanim.1	$$⊢Ⅎx\phi$$
Assertion	moanim	$$⊢{\exists }^{*}x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \to {\exists }^{*}x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		moanim.1	$$⊢Ⅎx\phi$$
2		ibar	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
3	1 2	mobid	$$⊢\phi \to \left({\exists }^{*}x\psi \leftrightarrow {\exists }^{*}x\left(\phi \wedge \psi \right)\right)$$
4		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
5	1 4	exlimi	$$⊢\exists x\left(\phi \wedge \psi \right)\to \phi$$
6	3 5	moanimlem	$$⊢{\exists }^{*}x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \to {\exists }^{*}x\psi \right)$$