Theorem mobidv

Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022)

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

Proof

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