Theorem rmoeq1

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023)

	Ref	Expression
Assertion	rmoeq1	$$⊢A=B\to \left({\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
2	1	anbi1d	$$⊢A=B\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \phi \right)\right)$$
3	2	mobidv	$$⊢A=B\to \left({\exists }^{*}x\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}x\left(x\in B\wedge \phi \right)\right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
5		df-rmo	$$⊢{\exists }^{*}x\in B\phi \leftrightarrow {\exists }^{*}x\left(x\in B\wedge \phi \right)$$
6	3 4 5	3bitr4g	$$⊢A=B\to \left({\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\in B\phi \right)$$