Theorem eubid

Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994) (Proof shortened by Wolf Lammen, 19-Feb-2023)

	Ref	Expression
Hypotheses	eubid.1	$$⊢Ⅎx\phi$$
	eubid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	eubid	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !x\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		eubid.1	$$⊢Ⅎx\phi$$
2		eubid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1 2	alrimi	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
4		eubi	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left(\exists !x\psi \leftrightarrow \exists !x\chi \right)$$
5	3 4	syl	$$⊢\phi \to \left(\exists !x\psi \leftrightarrow \exists !x\chi \right)$$