Theorem cbvrmow

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by NM, 16-Jun-2017) Avoid ax-10 , ax-13 . (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvrmow.1	$$⊢Ⅎy\phi$$
	cbvrmow.2	$$⊢Ⅎx\psi$$
	cbvrmow.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrmow	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvrmow.1	$$⊢Ⅎy\phi$$
2		cbvrmow.2	$$⊢Ⅎx\psi$$
3		cbvrmow.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4		nfv	$$⊢Ⅎyx\in A$$
5	4 1	nfan	$$⊢Ⅎy\left(x\in A\wedge \phi \right)$$
6		nfv	$$⊢Ⅎxy\in A$$
7	6 2	nfan	$$⊢Ⅎx\left(y\in A\wedge \psi \right)$$
8		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
9	8 3	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in A\wedge \psi \right)\right)$$
10	5 7 9	cbvmow	$$⊢{\exists }^{*}x\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}y\left(y\in A\wedge \psi \right)$$
11		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
12		df-rmo	$$⊢{\exists }^{*}y\in A\psi \leftrightarrow {\exists }^{*}y\left(y\in A\wedge \psi \right)$$
13	10 11 12	3bitr4i	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$