Theorem eubi

Description: Equivalence theorem for the unique existential quantifier. Theorem *14.271 in WhiteheadRussell p. 192. (Contributed by Andrew Salmon, 11-Jul-2011) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022)

	Ref	Expression
Assertion	eubi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\exists !x\phi \leftrightarrow \exists !x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exbi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\exists x\phi \leftrightarrow \exists x\psi \right)$$
2		mobi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left({\exists }^{*}x\phi \leftrightarrow {\exists }^{*}x\psi \right)$$
3	1 2	anbi12d	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\leftrightarrow \left(\exists x\psi \wedge {\exists }^{*}x\psi \right)\right)$$
4		df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
5		df-eu	$$⊢\exists !x\psi \leftrightarrow \left(\exists x\psi \wedge {\exists }^{*}x\psi \right)$$
6	3 4 5	3bitr4g	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\exists !x\phi \leftrightarrow \exists !x\psi \right)$$