Theorem rmoeq

Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018) (Revised by AV, 27-Oct-2020) (Proof shortened by NM, 29-Oct-2020)

	Ref	Expression
Assertion	rmoeq	$$⊢{\exists }^{*}x\in Bx=A$$

Proof

Step	Hyp	Ref	Expression
1		moeq	$$⊢{\exists }^{*}xx=A$$
2	1	moani	$$⊢{\exists }^{*}x\left(x\in B\wedge x=A\right)$$
3		df-rmo	$$⊢{\exists }^{*}x\in Bx=A\leftrightarrow {\exists }^{*}x\left(x\in B\wedge x=A\right)$$
4	2 3	mpbir	$$⊢{\exists }^{*}x\in Bx=A$$