Theorem mobid

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995) Remove dependency on ax-10 , ax-11 , ax-13 . (Revised by BJ, 14-Oct-2022) (Proof shortened by Wolf Lammen, 18-Feb-2023)

	Ref	Expression
Hypotheses	mobid.1	$$⊢Ⅎx\phi$$
	mobid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	mobid	$$⊢\phi \to \left({\exists }^{*}x\psi \leftrightarrow {\exists }^{*}x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		mobid.1	$$⊢Ⅎx\phi$$
2		mobid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1 2	alrimi	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
4		mobi	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left({\exists }^{*}x\psi \leftrightarrow {\exists }^{*}x\chi \right)$$
5	3 4	syl	$$⊢\phi \to \left({\exists }^{*}x\psi \leftrightarrow {\exists }^{*}x\chi \right)$$