Theorem eubidv

Description: Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022)

	Ref	Expression
Hypothesis	eubidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	eubidv	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		eubidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
3		eubi	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left(\exists !x\psi \leftrightarrow \exists !x\chi \right)$$
4	2 3	syl	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !x\chi \right)$$